Avogadro's number is pretty useful.

A googol is pretty useful, since it gives us a scale for reference. For instance, there are roughly 10^80 elementary particles in our universe. If we wanted a googol particles, we'd need a hundred billion billion universes all comparable to our own.

After that, I would say that most greater numbers serve no practical purpose, though it is still good to know them. For instance, Graham's Number serves as the upper bound to a solution to a particular problem in maths, with the lower bound being 6 or something. As one mathematician said, "There's room for improvement."

Factorials, while being exceedingly large (69! is just under a googol), are necessary for determining permutations and combinations.

There are exceptionally large prime numbers that have been found, on the order of millions of digits in length. They have some use in computer cryptography, and barring some advance in prime number theory and/or quantum computing, the ones we use right now are doing an exceptional job. For instance, RSA cryptography makes extensive use of near-primes, which are numbers that have 2 factors other than 1 and itself. The numbers used in the cryptography are massive, which makes finding their factors extremely difficult and time consuming. If we figured out algorithms to generate factors for near primes, then RSA cryptography would be rendered useless. Want to destroy the global economy, expose every classified secret on every database and expose everyone's personal information? Figure out the keys that are used in the cryptography or figure out an algorithm that can generate the keys for you and you'll be the most powerful person in the world (until someone kills you).