[binaryhermit]

Actually, last I saw, there was a limit on the number of characters in a domain, something like 56.

That said, you can use 0-9, a-z, and hyphens, so 37 characters.
37^56 is a ridiculous number. around 6.6 times 10^86.

EDIT: So, yes, finite, but with enough supply to be essentially infinite.

EDIT2: Apparently it's 63 not counting the TLD, but there's complicated rules regarding hyphens.

There apparently currently are 1584 TLDs, so the number of possible domains currently exceeds 1584*36^63 which is about 1.77*10^101. The high end of the estimate of the number of atoms in the observable universe is 10^82, so you could give every atom in the universe something along the lines of 10^19 domain names. So, technically finite, but yeah, infinite enough.