What is the maximum amount of cash banks typically allow for deposits?

Legally speaking, you can deposit whatever amount you wish, however, it will definitely be flagged (i.e. reported) and Philip Mikal gave the correct answer, it will be determined y the hold limit of the bank branch (essentially what the branch will hold in cash, as per the risks involved and insurance coverage).

Most large cash transactions are done directly in one of the main branches where vaulting services for large cash holds are provided for.