[Lord David]

Here's a list of potential aliens that could be seen in the new movie, without messing with canon or continuity issues.

Of course, not all species can be seen as part of Starfleet or whatever, but can be shown as background characters, visiting aliens or Federation Council members:

Based on first appearance: (*Note this is a list of Federation members by at the very least The Movie Era. There can easily be less members by during the mid 2200's)

ENT: (Essentially all could be Federation members, but are not represented in the Federation council)
Antarans
Arkonians
Axanar
Coridans (Definitely not a Federation Member as of the mid 2200's)
Denobulans
Enolians
Kantare
Kreetassans
Menk
Malurians
Mazarites
Retellians
Takret
Tandarans
Tarkaleans
Triannons
Vissians
Xyrillians

TOS:
Andorians (Federation members)
Ardanans (Federation members)
Argelians (Federation members)
Catullans (Probable Federation members)
Gideons
Kaylar (As Pike's first mission)
Klingons (TOS Era)
Orions
Tellarites (Federation members)
Tiburonian (Probable Federation members)
Tholians
Troglytes (Who are essentially Ardanans)
Vulcans (Duh) (Federation members)

TAS:
Caitians (Federation members)
Edosian (Probable Federation members)
Kzinti

MOVIE ERA:
Aaamazzarites (Federation members)
Arcadians (Federation members)
Arcturians (Federation members)
Ariolo (Federation members)
Arkenites (Federation members)
Betelgeusians (Federation members)
Bzzit Khaht (Federation members)
Deltans (Federation members)
Efrosians (Federation members)
K'normians (Federation members)
Kasheeta (Federation members)
Kazarites (Federation members)
Megarites (Federation members)
Rigellians (Federation members)
Rhaandarites (Federation members)
Saurian (Federation members)
Xelatians (Federation members)
Zaranites (Federation members)

TNG:
Bolians
Bynars
Kriosian
Nausicaans
Risans (Federation members)

DS9:
Trill (DS9 Look) (Probable Federation members)

Well that's what I can come up of for potential species. References from Memory Alpha.

No Gorns, please.

Hopefully we can see a Federation Council scene, show some of the species from the Voyage Home Council, other encountered species may not be Federation members by that point, so they can easily be visiting aliens or aliens encountered elsewhere.

Given the Council scene in The Voyage Home, it is possible that other known species may be Federation Members, but don't possess a seat in the council (Which would explain the 2 or more members of some species present in the council chamber). (Though I'd much prefer the council as being all species represented by a council member.

[Grand Admiral Thrawn]

I'd love to see more Caitians, personally. Maybe even M'Ress herself

If they do have Klingons, I wonder if they will have the head ridges. Maybe a mix of both looks.... (Yeah, ENT was good for something in giving us that explanation about the Klingon appearance change...)

[sfera]

cardassians!!!!!!!!!!!!!!!!!!!!

[Lord David]

Originally Posted by sfera

cardassians!!!!!!!!!!!!!!!!!!!!

Well Cardassians are a possibility, as Picard had encountered them during his time on the Stargazer, but I don't know, ideally I'd like them to stay away from that part of the Alpha Quadrant, as that would have meant potential contact with the Ferengi, who shouldn't be known by that point (Unless during this time the Ferengi went in a state of isolation, or it was the Cardassians who travelled near Federation space).

Barjorans for that matter are too a possibility.

However, I would like to see mostly TOS and TOS Movie Era Aliens.

[Indiana Jones]

I have to say that I wouldn't mind seeing a klingon or two, purely to see what JJ's take is on them.

[Workshed101]

I'd like to see more Deltans, preferably foxy chick Deltans

[Captain Hackett]

If Harry Ballz is here, I am sure he wants Mugato in the movie!

[Raiden]

I would just like to see how JJ tackles the whole klingon "forehead" debate

[HRH The KING]

Any of the following:

Andorians, Orions, Gorn, Klingons, Romulans.

That's enough for one movie.

[CoW mAn]

What's the lifespan of an Andorian? Could Shran still be around?