Please forgive what could be considered a slight topic drift here, but if we were to include the choices of Star Trek crews to include Trek novels, my favorite crew is that of the the USS Excalibur from the Star Trek: New Frontier series of novels. The series of ST:NF novels is long enough that you really do get to know the characters. And Peter David is great writer.
Captain Mackenzie Calhoun (rule breaker, authority bucker, warrior, purple eyes), Elizabeth Shelby (a crossover character from TNG from that Borg episode), Burgoyne 172 (a man/woman engineer), Dr. Selar (another crossover from TNG), Robin Lefler (crossover from TNG), Zak Kebron (he’s pretty much that rocklike guy in Fantastic Four), Mark McHenry (you find out his backstory late in the series, but it’s a doosey), Soleta (vulcan/romulan science officer), Si Cwan (royalty of a fallen empire–not part of the crew, but a key character), Night shift XO Kat Mueller, Morgan Primus (a triple crossover: she’s immortal, she’s Nurse Chappel, she’s Pike’s Number One and she’s Luxanna Troi). Oh, and even Arex and M’Ress from ST:Animated Series make an appearance.
I really recomend this series of Trek novels–entertainly–great character stuff, action and suspense and humor—and rich with references that trekkies will appreciate.
If I were just to consider the TV series crews, I guess I’d go with Kirk, Spock, McCoy and Scotty–and I mention just those four intentionally. Even though Chekov, Sulu and Uhura are a part of the Trek mythos, I don’t feel they ever got their fair share of lines or scenes.