I actually think that Khan is Kirk's greatest adversary.
Now, that doesn't mean this is the "best" or "my favorite." I'm thinking of it from a standpoint of pure impact.
1. Khan is engineered to be superior, intelligent, strong, arrogant. He is physically and mentally superior to Kirk, despite Kirk's ability to overcome that and get the best of him.
2. He was the first bad guy to hijack the Enterprise on the show, and if it wasn't for Marla's quick flip-flop, Kirk was moments from being killed in the decompression chamber, with Spock slotted to be next.
3. The events of TWOK are well known to everyone, but if you think about it...what happened in that film as a result of the conflict had more impact on the TOS universe than anything else.
-Khan crippled the Enterprise essentially beyond repair, sealing her fate
-Many crewmembers / cadets were killed or wounded. A member of Scotty's family died
-Spock is forced to sacrifice himself to allow the Enterprise to escape intact.
-The incident re-introduces Kirk to Carol and David, which ends up with the consequence of Kirk's son being executed by the Klingons
-Khan's theft and subsequent detonation of the Genesis Device re-intensifies the cold war with the Klingon Empire, results in a deadly skirmish, and a major intergalactic incident
-Kirk is demoted to Captain due to his actions associated with Spock's death and resurrection, and given a new starship command as a result.
-Kirk and company are uniquely positioned to save Earth from the whale probe due to the fall-out of all this.
I'd say no other adversary has had the impact on TOS or its characters than Khan has. For that reason, I'd have to say that I believe Khan is absolutely Kirk's "greatest adversary." That doesn't mean "most powerful" or "my favorite..." but from an impact standpoint, I don't believe anyone else even comes close. Khan blew the doors off the entire Trek universe at the time. That's saying something.