Do I need Hyperdimension Neptunia 2 and 3, when I have the first one? I see reviewers say, that they are basically the same game with slightly different features, like more player skins and clothes in the later games. Which of the games is the best one, if I want the best (and most complete) content? Or do I really need all three of them to fully enjoy?

You probably misunderstood something.
The gameplay and graphics are almost identical because die Rebirth-games are all remakes or the original trilogiy with the gameplay of the 3rd original one and were released within a small period of time.
Story is different for every game and not all of them have the same characters.
It's not like differnt versions of the same game.

The only review of the third game is a good example: https://www.gog.com/game/hyperdimension_neptunia_rebirth3_v_generation

Not sure, that I misunderstood this.

Some things are similar between the games (which is intended, often as a running gag through the entire series) but saying the games are basically the same is greatly exaggerated. To me is seems like the one who wrote the review was simply pissed about everything being reused aside from the added characters and story and wasn't a fan of said running gags and used the review as a exaggerated rant.

While dungeons / enemies etc are recycled throughout the series - the stories are different (and run together through the series - though each game can be played on its own without having played previous titles), new characters and new game mechanics are introduced throughout the series.

Try Re;Birth 1 - if you enjoy it get 2 & 3. If you don't you probably won't like the sequels.

Okay, I´ll try the classic approach. Thanks to all. :)

Good call

The battle mechanics are essentially the same in all 3 games (change a little re SP in 3 IIRC - though I may be mixing it up with VII) - but I have to say my favourite characters - the sisters... other CPUs / companions (no names to avoid spoilers if you continue on) get added in RB2/3/VII

and RB2 has *that* ending.... >_>