Correct spelling, explanation: the correct spelling is chef because the word comes directly from French chef (chef de cuisine. There is no double f there so obviously, it doesn’t make any sense to add such in English, especially that as an unvoiced consonant it doesn’t do anything to the word (no pronunciation changes, grammar changes, etc.). Therefore cheff is an incorrect form.

Definition of chef:
noun, the main cook, the person responsible for the food in a restaurant
– This steak was amazing. My compliments to the chef.
The local Italian restaurant is looking for a new chef.

Incorrect spelling, explanation: there is no such spelling as cheff. The word chef comes from the French chef, so the spelling hasn’t really changed in English. There is no grammar or orthography reason to spell it as cheff. Due to its origin chef is spelled with only one f. Pronunciation also doesn’t hint spelling it with a double f.

Incorrect spelling