I don't see your point. Also you're the one who started with semantics when you said that "immortality" means you can't make it stop. Which is why I asked you what term you would find better fitting to what I was proposing.
You are assyming that time is infinite, which we don't know.Infintie time means eventually, everything will happen(presumably, if you're truly immortal, excluding your death.)
Which means eventually a new universe will form around you.
Also if absolutely everything will happen (to me), that means I would go (among other things) through the most horrible experiences conceivable and inconceivable. Pass.