I don't think we ever saw Kirk and Co handle the Borg or Q. That could be interesting.
Technically, they did go up against a member of the Q continuum. They faced some omnipotent alien in ST:TOS who in later EU books was retconned into a Q.
I'd like to see the Gorn species show up. Have a modern recreation of that classic fight between Kirk and the rubber suited lizard man. And while they only appeared in ST: Voyager, I was always a fan of Species 8472: