You can NG+ as many times as you like so far as I know. All stats transfer, as do all all items(only enemy info key items tho the rest of the key items don't) and all weapons(ultima weapon and shield not included) and all accessories and garbs as well as progress on the Last ones quest though all other quests are reset. Your bestiary is also kept though if you change difficulty star rating is only attributed to the difficulty it was gained in but the kill count is set back to 0 so you can go out and kill all those monsters again. I think that's it.

Correct me if I am wrong but the ultimate dungeon can only be done on a new game + can't it?
It's just I did more than enough quests to unlock it and I just finished my first play through and it didn't unlock. :/

You're wrong. If you did at least 50 quests, it should have been unlocked. You would've had a scene and everything with an extra day for all the lost hours that were restored. It was in the Dead Dunes.

hmmm must have bugged then. Never mind I'm sure it will unlock this time through.

An exert from the piggy back guide to Lightning returns.

THE EXTRA DAY
Completeing the 5 main quests is enough to ensure you have access to the final day at the end of day 12 and the opportunity to see the story to it's conclusion. However if you have completed a sufficient nimber of sidequests by the end of day 12 you will unlick a special 14th day occurring between the 12th and final day. The exact formula used is too complicated to detail here but it's enough to say that each quest is allocated a value based on it's difficulty: 1star=0.02, 2star=0.04, 3star=0.08. The extra day is unlocked if you reach a value of 1.00 though the calculation is not a simple addition and leads to different results depending on the order in which you complete quests. Canvas quests do not contribute to this total.

I figured with your last comment this might be useful to somebody.
By the way I went and looked at an old save on my PS3 version and you seem to be right about the ultimate lair being able to be accessed on a first playthrough. I must have bugged somehow. :/

Moved thread to FFXIII-LR Subforum.