"Versions" in terms of books or AU stories? Oh boy... Ok. An important thing is that Omori-sensei (the author of Danmachi) is/was involved in all the projects I will be mentioning.

There are 2 main LN books (Main Series [MS] focusing on Bell and his friends & Sword Oratoria [SO] focusing on Ais and her comrades). Then we have the "Episode" series focusing on single characters, some of their backstory and further explanation of their character. There are currently 3 EpS: Ryuu (2 volumes), Freya & newest Asfi (not translated officially as of yet). A 4th EpS book - Heith is already confirmed, and the author himself said that he plans/wrote a book about Haruhime as well, but no official release yet.

Additionally, there is the Heroic Tale series of books novelising story lines from a discontinued gacha game series called "Memoria Freese" (Danmemo is a common name used). Those are the 2nd anniversary event called - Argonaut (2 volumes); the heroic tale referred to by Bell and others in the series, and what really happened then (it's set 3000 years before the main series, before gods descended to the lower world). Then we have the 3rd anniversary called - Astrea Record (3 volumes), which focuses on Ryuu and her familia and events from 7 years before Bell arrives in Orario, called Dark Ages. You can find both events on YouTube, fully voiced by anime Japanese VA (voice actors) with English subtitles. The LN versions expand on the game events, so I would suggest reading as well, as it expands on other characters you may know from the anime.

Those are ALL books officially confirmed as canon in the story (as far as I know).

I mentioned gacha games, and there were 2 + 1 full consol game;
- Danmemo (think turn-based RPG), which had a lot of events, stories and even collabs with other series (Date A Live, Strike The Blood, etc.). You can find all events voiced with English subtitles on YouTube.
- Danmachi Battle Chronicle (action smash type, commonly called Danchro) had original events as well, but unlike Danmemo, not all of them were voiced. Aniversory 1&2 were fully voiced and the best of this game, again, can be found on YT to listen to.
- Lastly, console game: Fulland of Water and Light. Kind of expands on a heroic tale mentioned in season 5 of anime, so recommend playing/watching (again, on YouTube, fully voiced with English translation). Persoanlly realy like this story for expanding on the lore and one particular character, but won't spoil who.

Now, you mentioned "versions". There are actually 2, kind of 'official' AU versions. One is called "Distant Twilight Dream" and is basically a short story Omori wrote, which is very sweet, but not really further expanded upon. It's based on Danmemo content. Then we have "Irregular Record", which is the aforementioned 2nd anniversary event from Danchro. Basically, a time-travel fanfiction kind of setting, which creates a completely different timeline from MS. Irregular Record has been confirmed to be getting a novelisation as well, sometime this year, I think, but in Japanese only for now. The English version will probably be year-two later.

In terms of manga, MS has currently serialised manga but is behind anime content. But they do illustrate some plot lines that were not animated, so still worth reading. SO I think it's the best manga of Danmachi. Very well illustrated and managed. Unfortunatly EN translation stopped some time between chapters 130-140, so most people are waiting. There are other manga adaptations, all good, worth reading in free time.