e comes from evaluating the expression (1 + 1/n)^n
As n gets larger and larger, it starts to get closer and closer to the value of e (which truly can't be reached exactly, since e is an irrational number).
Simple table showing what I am talking about:
n (1 + 1/n)^n
100 2.70481
1000 2.71692
100000 2.71827
1000000 2.71828
and so on...
e is used quite a bit in calculus and comes up in mathematical theory and in applications (pretty much anything that uses calculus as a base, i.e. engineering and physics).
As far as the half the distance thing mentioned by Grim:
Yes, in theory, you won't reach the object. But I haven't ever met a mathematician that says you would never actually reach it. The half the distance theory is only truly applicable when both the thing moving and the destination occupy no space. Basically, it means the two things are points.
But, as you and everyone else are aware, this is not the case in the real world. It is an interesting topic of discussion in math classes, but that is about it.
