If numbers never end, then how do they all have names?

Not all numbers have names. Very larger numbers are expressed by just writing out their digits.

There is no last number. As you can think up the biggest of the biggest (the equivalent of lightyears) I'll add a one. -- and another one and another one and then I'll multiply them by themselves.

Peace.

Actually we don't have names for all of the numbers because there are infinitely many of them. The numbers are more than just the integers, there are fractions and irrational numbers and we haven't given them all names.

Even if you restrict yourself to the whole numbers (zero, one, two, three, ..., million, billion, trillion, ...) we don't bother giving them English names after about a million digits. At that point it is easier just to spell out their digits and/or use scientific notation.

This is almost as foolish as asking how we could have so many words when we only have 26 letters to work with.

Perhaps you could say we know how to name any number uniquely when a particular number needs to be addressed. And bear in mind they don't tangibly exist other than as a useful idea in our brains.