Good Lord, I'm old, apparently. Also, my taste in Trek games is very subjective, and what I enjoyed about them may or may not have much to do with why you would or wouldn't, honestly.
My top 10 Trek games:
10. Star Trek: Strategic Operations Simulator. The arcade machine from 1982. I got it as a cartridge for my Atari 400 computer in 1984. I've loaded it up in an emulator recently. It wasn't that great - basically, just a very simple move-to-the-right-place shoot and don't get shot sort of thing. But it certainly seemed awesome back then. I think mostly it was the Trek theme playing and the fact that they got some of the sound effects (like Red Alert) right.
9. Space Quest V and VI. PC. So you like the worm, huh? So you want to ride the worm, huh? I LET YOU RIDE THE WORM!

Not actual Trek games, per se, but both games featured PLENTY of Trek parody. V put you in the captain's chair of your own ship. And VI was, well, VI. A lot of it defies simple explanation, but, well, here:
Just one tiny part of what made it great.
8. Star Fleet I: The War Begins! - Apple, Commodore, Atari, PC. This was basically an expanded version of EGA Trek released in 1984. Great music (for MIDI), a great manual, a bit of story added, and of course, Klingons were "Krellans" and Romulans were "Zaldrons", because it didn't have an official Trek license.
7. A Final Unity. This would probably be higher if it hadn't been such a pain in the ass to run and to work around glitches. But overall, a great game, even so.
6. Star Trek Online. I really, really enjoyed the game City of Heroes before it was cancelled, and this is made by the same studio and is very similar in how it works. I'm a little disappointed that I can't fly like I could in CoH, but that wouldn't be very Trek, I don't guess, so... I also get a little annoyed with how railroad-y missions are. But still, there's so much neat Trek stuff to see and play with, and I can sometimes play with my son, who isn't at home anymore.
5. Star Trek 25th Anniversary / Judgement Rites. This is on everyone's list, somewhere. No need to explain.
4. Star Trek: The Kobayashi Alternative and Star Trek: The Promethean Prophecy. 1985 and 1986, PC, Amiga, Apple, Commodore. Text-based but with an unusual interface for a text game that worked really well to allow conversations with crew, ran in under 512k, still the best simulation of life as Captain Kirk I've seen in any game.
3. FASA's Federation Ship Recognition Manual. Yes, I'm aware this was only one part of FASA's Star Trek RPG. I don't care - it was the only part of it I cared about, since I didn't have anyone to play the game with. It sparked my love of drawing starships as I would copy the pictures, and it contributed heavily to developing my love of Trek in general.
2. Armada/Armada II - played this on networked PCs with my son when he was growing up.
1. A homebrew Star Trek / Star Wars / BSG / and whatever else we felt like pencil and paper RPG that my wife, three of my best friends, the mom and sister of one of them, used to play and I used to run using d20s, d6s, and half-remembered, half-made-up 2nd edition D&D rules and a lot of Rule 0. Absolutely the best. Their ship and the ship they were fighting were disabled and racing to see who would get power back to weapons and propulsion first. My friend's sister was playing a Data-like android, and she had the Wookie (played by another friend) *throw* her out of the airlock at the other ship, so she could go finish them off. Good times.
