Off the top of my head the ones involving the Borg and the Xindi (Xindi took up about a season or so of episodes I believe). Which is fine with me, Enterprise suffered from trying to have all the big name Star Trek races even when it did not make sense for them to be there. Though it had some nice ideas as well.
Also I miss the Trill though DS9 is my favorite of the Treks so Dax was a cool character and we got to see a lot of the race through her instead of being a one off race. I think the biggest issue is most races in Trek never got much screen time so did not develop a character. The Bajorans got it thanks to DS9, though Ro from TNG helped there, Betazoid kind of had it with Troi (both of them). Klingons and Romulans were antagonists that became recurring along with the Cardassians and Ferengi so they had their cultures developed and fleshed out. Otherwise expies from other sci fi modified to fit the Trek feel works best since there is more to draw on.
So for me races I want to see: Trill (liked the Dax character and the symbiot that provides an extended time frame can really make things interesting since they may have witnessed events that took place well before the rest of the crew was born but we also see in the Jazdia to Ezri transition that the host brings their own perception to the mix and in many ways allows them to re-examine past memories in a new light.)
Ferengi- major race that is not necessarily antagonistic but can be and provide both completion and I think could be key to having a non-aligned bloc thanks to their economic power along with a solid military (Marauders were no joke even for a Galaxy Class).
Gorn- they are a personal favorite of mine both from the episode in TOS and from Starfleet Command 2, I loved playing their big tough ships with plasma torps. They also provide a sort of peer power that would not be an affiliate which keeps things interesting. And for that matter the Ferengi provides a power that has no interest in being anyone's servants but also is big enough that you have to engage with them.
Breen- from DS9 they show a certain level of strength to be independent in their own right so them showing up at some point as another power (maybe bigger than the Apiata but smaller than the Romulans?)
Just having multiple powers like the Breen and Gorn and Ferengi can shake up the diplomatic and power balance without needing a group as big as the Federation. Shifting alliances and diplomatic relations between multiple factions would make for an interesting peace time. Are we trying to stop a war between the Breen and the Romulans? Is one side trying to pull us in? Are the Klingons and Breen making common cause? Just having the larger species all trying to influence smaller ones and having different relationships prevents the need from having species X expand to be a threat.
After all it wasn't just the Cardassians but the Dominion as well that gave them the power to oppose the Federation and Klingons, and then when the Romulans joined to tip the balance the Breen joined later to tip it back. So with just 4 major races or blocks it either is a 2v1 or 2v2 situation or 1v1v1v1, put in a 5th race or 6th race with their own interests and the situation changes.