I've always adhered to the idea that "infinity" encodes "allness". For instance, to say that the sum of 1/2^n for all natural numbers n converges to 1 is not to say that we're actually adding up infinitely many numbers, but rather that I can always win a certain game: you give me an arbitrarily small epsilon > 0, I can give you enough terms in the sequence such that their sum (a finite sum) is within epsilon of 1. No, I can't actually add up infinitely many numbers, but you can never win my game, so certainly "infinity" exists in that sense.

So, while I can't "point" to an infinite number of things like I can point to 9 things or 3.62 things, I still think it exists.

I'm not sure how well this generalizes to all infinite cardinals, ordinals, or to transfinite induction/construction. It is certainly strange that Cantor's theorem (the cardinality of a set is strictly smaller than that of its power set) implies there are different sizes of "all" implicit in my usage of the word.