As far as I know, you’re right. During the touching scene at the end of Generations between Data and Spot, I was going, “Wait … what about Livingston? Where’s Livingston?! What happened to him?!?”
Yep. Poor guy. The character’s name was Joachim, and the actor who played him was Judson Earney Scott – who, by the way, was the same actor who played “Bennu” in the short-lived television series The Phoenix (a series that followed the same basic formula as Kung Fu and The Incredible Hulk).
Some with lines, some without.
–The Latino-looking fellow standing by a grinning Scotty as the Enterprise leaves spacedock in “The Undiscovered Country.” The poor guy’s a big-time mouth-breather and he is staring forward in a dazed, retarded manner.
–That hot freckled redheaded assistant to Geordi who gets Borgified in “First Contact.”
–The Latino Com Offier from “Search For Spock” who says something like: “This is U. S. S. Grissom calling on newly-installed super-secret, unbreakable 2 million megahertz channel 97.5 & 1/2-D.”
– Disco Dr. McCoy’s mo-hunking necklace from “The Motion Picture” (and that beard, too!).
–The lone Klingon who dared laugh at McCoy’s “touch of arthritis” joke in “The Undiscovered Country.”
–Punk On Bus (How many megatons… I say “Screw You,” etc)
Sir Rhosis
From a close examination of the credits the guy playing the punk is apparently also the guy you can hear singing on the tape.
Ooooh, ooooh, another one from that movie: “Did you see that?” “No, and neither did you!”
Punk on Bus was played by one of Nimoy’s assistants, Associate Producer Kirk Thatcher, also a musician. He is now a middle-aged, balding, overweight middle-class looking fellow.
Don’t know what the two garbagemen are up to. Do people still use toaster ovens?
Other candidates for regular status on the proposed show:
–“Make-shift soLAR sail” Captain (Vijay Singh) from “The Voyage Home”
–The cute little Go-Go (Weidlin?) on the background screen in “The Voyage Home”
–The African-America cop and his Greenwich Village mustache who arrested McCoy in “The Search For Spock.”
–The Navy Commander who says something like: Comand Deck, this is Command Deck Commander, Commander Rogerson" from TVH.
Sir Rhosis
It’s all about that poor dog they dressed up as some kind of alien in the TOS episode where Kirk gets split in two in a (go figure) trasporter incident.
Those darling little baby Xindi Insectoids on Enterprise.
I hate to break it to you…but you remember the Seventh Season episode, Genesis?
[spoiler]…Data and Picard discover caveman-Riker in the Ready Room, rummaging through Livingston’s broken tank. I’m afraid he might have been fish sticks before Generations even happened. (Which it didn’t, as a matter of fact. It never happened.)
Of course, the Virus affected the animals on the ship, as well as the crew. For all we know, Livingston had just mutated into a dinosauroid, busted out of his own tank, and was makin’ it with mutant-Troi by the time Data and Picard got back.
[/spoiler]
Four television series. Enterprise doesn’t qualify. 
Rom. One time, Quark said, “Rom is an idiot. He couldn’t fix a straw if it was bent.” Man that’s gotta stink… even the guy’s character is unappreciated. IIRC he did speak a few times, in a voice intended to portray someone of lower intelligence.
Rom became a major recurring character during the middle and end years of DS9 and was the Grand Nagus when the show went off the air.
I don’t think he counts. 
Yeah, I wasn’t talking about Rom. There’s definitely another Ferengi waiter. Now that I think about it, I think he may have had a speaking part in the episode where Quark’s employees go on strike.
You guys have come up with some great ones, though. I had forgotten about the Klingon with the blue food in his mouth and Spot. How could I forget Spot? That part was played by about a dozen different cats, and each one brought their own unique interpretation to the role.
Also, I forgot another one of my favorites: Normal-Eared Tuvok from when he played a different part in The Next Generation show and the Generations movie. He must have gotten them caught in a rice picker or something to account for his later appearance.
“Captain! Over here…I’ve found something…Yeeeaaaarrrgh!”
…
The next time you watch Star Trek:The Motion Picture, check out the guy Spock neck-pinches on his way to nicking a spacesuit to go outside the ship and commune with V’Ger. Longest, most drawn-out, overdramatized reaction to the neck-pinch ever.
Kirk: “Five to beam up, Scotty.”
Scott [in total shock]: “Yeh only lost one Security officer, Captain?”
In the first ST movie, Kirk is on the Enterprise for the first time in years, and crew members are scurrying around trying to get everything ready for launch. There’s one chick who whines “What is the PROBLEMMMMM? Why wasn’t this fixed an HOUR AGOOOOOO?”
The no-name officers whose sole duty seems to be to keep the navigation seats warm and leave when the primary cast members show up.
Christian Slater as Sulu’s bitch.
How about we have a series set on Risa?
You know it would be something along the lines of Newheart. Just an ordinary ‘guy’ with a wacky staff trying to run a space brothel.
My favorite is Beauregard, Sulu’s pet plant from the episode where Kirk gets split into Good Guy and Bad Guy by the transporter. Sulu introduces the plant to - who? Yeoman Rand? Uhura? One of the regular hotties - and explains how into the plant he is and how it reacts to negative emotions…it’s exclusive use was to identify Bad Guy Kirk at a point in the plot.
I remember as a Young Star Trek Geek watching that episode and remembering that plant’s name figuring it would matter in later ep’s, too. Never happened.