My intuition about infinite time used to go something like this:
every possible event will happen, if you wait long enough.
Consider a random walk on a 2-D grid. Will the drunkard return to
the starting point? Yes, if you wait long enough: you can prove
that the random walk will, eventually, return. (Formally, it
returns "almost surely" or "with probability 1". For example, the
random walk could proceed due east forever and never return; it's
not impossible, just infinitely improbable.)
In fact, you can prove the same thing about every other point on
the grid, not just the start. Each point will ("almost surely") be
visited during the random walk. If you wait long enough, the
drunkard will wander everywhere. Score a point for the "everything
happens eventually" idea.
But that intuition is wrong. Forever is not always enough.
Move the drunkard to a 3-D grid. Now the probability of returning
to the start is not "almost sure". Instead, it's a little over
one-third. If you start three drunkards on a 3-D random walk, you
would expect only one of them to ever come back, even if you wait
forever. And the odds are even worse with more dimensions.
Some events still don't happen after infinite time. Infinity isn't
as long as you think.