We see the phrase "almost infinite" (or "nearly infinite", or "infinite for all practical purposes") in much literature for the layman, usually to describe a vastly large number of combinations or possibilities from a relatively large number of items. For example, all of the possible brain states for a human brain (comprising about 3 billion neurons), or all possible combinations of a million Lego blocks, etc.
Obviously, these are in actuality just large finite numbers; having an infinite number of permutations of a set of objects would require the set to be infinite itself, or the number of possible states of each element would have to be infinite. Most uses of the term "infinite possibilities" or "almost infinite" are, in fact, just large finite numbers. All of which are, of course, less than infinity.
But is there some mathematically meaningful definition of "almost infinite"? If we say that m is a "nearly infinite" number, where m < omega, but with m having some property that in general makes it larger than "almost all" finite n?
Personally, I don't think there is such a definition; but then I would enjoy being proved wrong.