Aside from English and possibly Russian (thanks to Ivanova), I can think of 6 for sure, and possibly 3 more:
- Minbari, as mentioned in the question.
- The Gaim, who are insectoid and can't make English sounds - all the Gaim ambassadors carry translation devices, and I believe you can hear their language for a moment before the translation is heard.
- The Pak'Ma'Ra, who may or may not be able to make English sounds. Their religion might play a part in it. They use a similar translation device as the Gaim.
- The Vorlons, who may or may not be able to make English sounds. If they can, they've chosen to never do so. The musical noise you hear just before and during when they speak is their true voice, what you hear in English is their translation device.
- The Shadows, whose speech is, as far as I recall, never translated. You can hear a sort of chittering noise when Mr. Morden is speaking to them. (It's sort-of translated in the final battle, but I think that was mind-to-mind communication, not actually translating their speech.)
- One of the unnamed First Ones, but only for a short part in one episode. The one that manifested like a Moai head refused to speak English, although it understood the language.
Babylon 5 largely makes use of Translation Convention (TVTropes) where if, say, Centauri are speaking to Centauri with no other species around, they'll be speaking Centauri and we'll be hearing English.
Because of this, Centauri and Narn are the two I'm uncertain about. They play such a large role that it seems strange that we'd never hear their language, however the characters we know best have learned English so they can get by on the station, and the others are generally only seen with their own kind. If we hear their language at all, it's most likely 1) when they're swearing, 2) during a Narn religious ceremony, 3) later in the series when G'Kar is writing his book, or 4) when he tries to share the Book of G'Quan and Garibaldi fails to translate it.
I also don't quite recall if we've ever heard the Drakh language. Like the Shadows, they were meant to be mysterious, but they used Minbari in ship-to-ship communication when they were first introduced. Most likely, if we heard their language, it was either on Centauri Prime or in the Crusade spinoff.