Is there a biggest infinity?

No. For any infinity you can name, there is a strictly larger one, so there is no largest infinity. That is Cantor's theorem, first stated and proved at the end of the 19th century by Georg Cantor.

The recipe for climbing is simple. Take any set, and form the set of all its subsets, called its power set. Cantor showed that the power set always has strictly greater cardinality than the set you started with. His argument applies to any set, finite or infinite. Apply it to the whole numbers, and you get a set that is strictly larger: the real numbers have the cardinality of the power set of the integers, which is strictly larger than the cardinality of the integers. Now take the power set again, and again, and each time you get an infinity strictly larger than the one before. The result is an endless hierarchy of infinite cardinals.

The proof is elegant and remarkably simple, and it is a diagonal argument in the same spirit as the one that shows the reals cannot be listed. Suppose you could pair every number with a set of numbers, with nothing left over. Some numbers would be paired with a set that contains them, and Cantor's proof calls these selfish. Others would be paired with a set that does not, and the proof calls these non-selfish. Now gather all the non-selfish numbers into one special set. That set provides the contradiction: no number can be its partner, so the pairing was never complete.

Cantor gave essentially this proof in 1891. It had immediate and important consequences for the philosophy of mathematics, and it also led to a famous problem: if you try to gather everything into one set containing all sets, Cantor's theorem says its power set would be strictly bigger, yet a set of all sets would already contain every one of those subsets. That contradiction is called Cantor's paradox.