How many electron orbitals exist?

Technically, an infinite number of atomic orbitals exist, but we really only care about the ones that atoms actually use. To date, the orbital shape of the highest $l$ in use is the $5 f$ orbital.
So, for general intents and purposes, we really only need to understand the $n s$, $n p$, $\left(n - 1\right) d$, and sometimes the $\left(n - 2\right) f$ orbitals. I don't anticipate chemistry education to need to incorporate the $\left(n - 3\right) g$ orbitals into practical applications for years.