No Feats. Skills?

[Chalkarts]

One thing I've struggled with in 5th is that the only way to get new skills is from the Skilled feat.

I like Skillmonkeys.

If I were to play in a game that didn't use feats, would I be unable to learn any skills beyond those gained at creation?

[Trandir]

You can gain some via multiclassing, ranger, rogue and bard provide 1 extra proficiency, and some subclasses grant extra proficiencies as well (scout or college of lore to say 2)

[Chugger]

and Knowledge Cleric

[Chalkarts]

You can gain some via multiclassing, ranger, rogue and bard provide 1 extra proficiency, and some subclasses grant extra proficiencies as well (scout or college of lore to say 2)

I could be a Rogue that dips 1 lvl from each class that will give him a new skill.

[Cheesegear]

If I were to play in a game that didn't use feats, would I be unable to learn any skills beyond those gained at creation?

Skillmonkey Build

Background - 2 Skills
V. Human - 1 Skill
Skilled Feat - +3 Skills

Cleric (Knowledge) 2 - Two skills and an another Knowledge skill. Knowledge of the Ages
Ranger 1 - +1 Skill
Bard (Lore) 10+ - +1 Skill, Jack of All Trades, Expertise (x4), +3 Skills.
Rogue (Scout) 6+ - +1 Skill, Knowledge (Nature) and Survival, Expertise (x4)

At some point, if you have 4 levels in your class, pick up Prodigy (Xanathar's); +1 Skill, more expertise.

With proper planning, by level 19 you can have proficiency in all 18 Skills, and expertise in like, 9?

[Lvl 2 Expert]

I could be a Rogue that dips 1 lvl from each class that will give him a new skill.

If you take 3 levels of bard you'll also gain jack of all traits and expertise, as well as the tree extra skills from lore bard. If you make sure the ranger level is revised ranger you get advantage on initiative rolls as a nice bonus, although to get both of these classes you will need to invest in a 13 in both charisma and wisdom.

Your sneak attack damage will be 4 levels behind at this point, pretty much completely down the drain. But you will have skills, 4 (rogue)+1 (ranger) +1 (bard) +3 (lore bard)+2 (background) + usually 1 or 2 for your race = 12 or 13 skills, out of what, 18 total skills available? (Plus thieves tools and a musical instrument and 2 tools/languages from a background.) And the skills you don't have get jack of all trades.

Then as Cheesegear has posted while I have this open you can get another 2 skills (and two languages) for 2 levels of knowledge cleric, since you already had to get 13 wisdom for going ranger anyway. So that brings the total to 15 out of all 18 skills with 6 levels of dip on your rogue, assuming a race that gets two skills, like Eberron Orcs. What? Orcs make great rogues. Find a weaponsmith that can craft a custom finesse greataxe and you're good to go. If you were allowed feats you could get the skilled feat at level 4 and have all skills by level 7. Four of those skills benefit from expertise at that point, and with another 5 levels of rogue that number goes up to 6.

Alternatively/additionally, you could follow through on the ranger to level 3 and go beastmaster. Your companion gains proficiency in two skills, and some of the companion options already have 2 skills to begin with, for a total of 4 extra skills. Two of these skills will probably be stealth and perception (with advantage for either smell of sight), or some variation on that theme, but the other two can be skills for one of your lower stats (assuming the animal has a better one). So using an animal companion you can actual be a 9th level character that without any feats has all skills (between herself and her companion), and an extra skill slot that they don't know what to do with. Take perception on both of them for sort of double-advantage maybe? At this point you can also choose to start as a ranger. It will give you one skill less than starting as a rogue, but you might enjoy getting the medium armor and martial weapons since one skill less still means you get all the skills. Add five more levels of rogue for a level 14 build with all skills and expertise in 6 of them. (But none of those expertise skills can be the skills on your companion, who by the way scales with ranger level so is pretty useless in a fight at this point.)

Would be kind of a weird build, ranger 1/rogue 1/lore bard 3/ranger +2/knowledge cleric 2/rogue +5/??? (+)6
(In fact, maybe go ranger 3 first, since your companion is the one getting perception. A skill monkey that's basically blind until level 7 is kind of weird.)

EDIT: Oh wait, you only need one level of knowledge cleric. Second level is only for chumps who don't have all proficiencies already. So you can without feats actually have all skill proficiencies by level 8.
As an additional afterthought on using an animal companion, the book companion feature states that "your companion uses your proficiency bonus rather than its own." So you don't need ranger levels to get good skills on the beast. As long as the HP and attack of the companion don't matter to you.

[Cheesegear]

Fixed Skill Progression

Background; Entertainer: Acrobatics, Performance (2)
Variant Human: Athletics (3)
V.Human Feat; Skilled: Persuasion, Sleight of Hand, Stealth (6)

Cleric 2 (Knowledge): Insight, Medicine, Arcana, Religion (10)

Ranger 1: Perception (11)

Bard 3 (Lore): History, Investigation, Animal Handling, Deception (15)

Rogue 3 (Scout): Intimidation, Nature, Survival (18)

You could change some things around. But the point is, that it can be done. You are now proficient in everything.

Starting off as Cleric is important, 'cause you can't multi-class into Cleric and get proficiency. You then pick up Knowledge of the Ages so you can become proficient in anything, whilst you aren't. Next multi-class into Bard, to pick up Jack of all Trades at 2, so even when you aren't proficient in something, you are. And then continuing to 3 for Lore for three more skills. You're really shooting for Bard 10, Rogue 6. But it doesn't matter what order you take them in, since you're only doing it for Expertise. You have every Skill by Level 9, if you want.

After Level 5 (Cleric 2/Bard 3), the order you take the rest of your levels in will probably be determined by your party and the kinds of adventures you're on.
But certainly, between Knowledge of the Ages and Jack of All Trades, you can already almost do anything you want.

[OldTrees1]

Starting off as Cleric is important, 'cause you can't multi-class into Cleric and get proficiency.

If you start as Rogue you get +2 skills (4 instead of 2).
If you multiclass into Rogue you only get +1 skill.
Start as Rogue.

Rogue 1 / Knowledge Cleric 1 = 4+2+2 skills
Knowledge Cleric 1 / Rogue 1 = 2+2+2+1 skills
8 > 7

[Cheesegear]

If you start as Rogue you get +2 skills (4 instead of 2).
If you multiclass into Rogue you only get +1 skill.
Start as Rogue.

Rogue 1 / Knowledge Cleric 1 = 4+2+2 skills
Knowledge Cleric 1 / Rogue 1 = 2+2+2+1 skills
8 > 7

I mean, my way, you end up being proficient in all 18 Skills anyway, and your party starts the first two levels of their adventure with a Cleric, which is always nice at lower levels.

But, you're right. If Feats aren't allowed (as per the Title/OP), then I doubt V.Human is legal (-4 skills)

Half-Elves give two free Skills. Starting with Rogue 1, instead of Cleric 1, gives you +1 Skill (from a set, but you should still have many more 'free skills' to choose from to cover your bases).
You are now only short a single skill. I'll put forward that it's not necessary for one character to be proficient in all three 'talking skills'.

[OldTrees1]

I mean, my way, you end up being proficient in all 18 Skills anyway, and your party starts the first two levels of their adventure with a Cleric, which is always nice at lower levels.

That is true. Although saving 1 proficiency means you can reclaim 1 of the multiclasses (no ranger level?).

However I agree Cleric is nice. I had a dungeon rogue that started Knowledge Cleric 1 / Rogue X.

[Cheesegear]

That is true. Although saving 1 proficiency means you can reclaim 1 of the multiclasses (no ranger level?).

If the Feat is allowed (it isn't), I think it's better to go V.Human starting with Cleric 1.
However, because Feats aren't allowed, Half-Elf Rogue 1 is needed to make up the difference.

I'm pretty sure, at the very least, Cleric 2 (Knowledge) / Bard 10 (Lore) is a viable character concept.

[Biffoniacus_Furiou]

PHB p187, the last thing on the page is training during downtime to learn new language or tool proficiencies. Not exactly skills, but given how Xanathar's expanded on what each tool proficiency allows you to do, those are almost as good as skills. With sufficient time and gold there's no realistic limit to how many extra proficiencies you can gain.

Tasha's is introducing swapping martial weapon and tool proficiencies for each other, so that may also expand on ways to gain new skills.

[Ashrym]

Fighter subclasses and warlock invocations are also sources for skills.

The character should gain reliable talent to really excel.

[sithlordnergal]

So many build options, and they all use the Skilled feat...when the OP specifically said "No feats". Well, let me help you out:

Race: Half-Elf

Classes: Lore Bard / Great Old One Warlock / Knowledge cleric


Str: 8
Dex: 14
Con: 14
Int: 10
Wis: 14
Cha: 16


You'll want to play as a Half-Elf because it gives you two skill proficencies of your choice. With it, Bard, and your Background, you'll start off knowing 7 skills of your choice with one set of tools. At level 3 you can take College of Lore to get another 3 skill proficiencies, bringing you up to 10 skill proficiencies of your choice.

Next you'll want two levels of Warlock. I'd take Great Old One simply because it lets you speak all languages with the Awakened Mind feature. But more to the point, at level 2 you get Invocations, and one Invocation called Beguiling Influence gives you proficiency in Persuasion and Deception. Now you have 12 skill proficiencies.

Finally take at least one level of Knowledge Cleric, this will net you two Intelligence based skill proficiencies of your choice and you can use Channel Divinity to get proficiency in one skill/tool of your choice for 10 minutes.

In total, you'll have 14 static skill proficiencies, with one skill/tool proficiency that you can use for anything you might not know. You'll also be able to essentially talk to any and all creatures thanks to Awakened Mind, giving you a form of Language proficiency. Given there are only 18 skills on a character sheet, you'll only be missing 4 of them, and even then you can give yourself proficiency in one of those four remaining skills with your Channel Divinity at any time.

EDIT: Not to mention you'll add half of your proficiency modifier to those four skills that you don't know, making you better at them then you normally would be. That said, you'll need to be level 6 at a minimum to get all 14 skill proficiencies since you need 3 levels of Bard, 2 levels of Warlock, and 1 level of Cleric. And if you end up in a game with Feats, you can take Skilled and Prodigy to get proficiency in every single skill with this build.


EDIT

So, for an easier build you can just go

Race: Half-Elf

Class: Scout Rogue/Lore Bard

If you take Skilled and Prodigy you gain proficiency in every single skill, and its much easier then my first build. Scout Rogue and Lore Bard will net you 10 skills, Half Elf gets you up to 12, Background gets you to 14, then Skilled and Prodigy give you the last 4 skill proficiencies.

[Lvl 2 Expert]

So many build options, and they all use the Skilled feat...

No I didn't.

I mentioned it, like you do, (maybe a bit less than you do,) but I presented a full build with all skills using no feats at all.

Was it a practical build? Hey, look at the time, I need to be going.

[Ashrym]

Half-elf, start rogue. 8 skill proficiencies. Take at least 11 levels for reliable talent. Go arcane trickster to add caster levels.

Splash bard. Go lore. 4 skill proficiencies.

12 skill proficiencies plus reliable talent and jack-of-all-trades plus expertise should be plenty for a skill monkey without feats. You can add more.

Splash warlock. Take at least 2 levels for 2 skill proficiencies via invocation. Less MAD this way too.

And/or splash knowledge cleric for 2 more plus the channel divinity feature. Adds MAD.

And/or splash 3 levels of fighter to grab a subclass proficiency.

Alternatively, half-elf rogue 11, artificer 7, and bard 2 gives 9 proficiencies and jack-of-all-trades plus tool expertise plus flash of genius. Also adds MAD.

The bigger issue is delayed abilities from MC.