The title says it all. My picks are as follows.
1. The Infinite Dungeon. The thought of a place that goes on forever as far as our real-world lifespans are concerned, where you can never revisit a particular level, has always fascinated me. Also, it is a good place where to gain levels and get items as most of us know, plus the feeling of acquiring gear as you go adds to the excitement of the game. I've had characters that entered with their starting gear, and came out with 20 PV, holy water, useful items, numerous intrinsics (you might shout "grinding", but I enjoyed the task).
If you've won the game, you can always try to see how deep you can dive in the ID - the fun of dungeon crawling continues with no background corruption and ever-increasing difficulty. Props to those who have gone beyond I300, I'd like to do something similar one day.
2. Darkforge. I generally tend to have horrible luck with this repository of high-DL weapons and armor, but you never know what you're going to get - dragon scale mail, eternium two-handed swords, Executor, weapons of devastation, knives of endurance, etc. There's always one item that's very useful, however, at least for my wizards: the crown of science. You can stock on books, wear it in the wilderness for its ludicrous Le boost, read everything, then uncurse and remove.
I thought the only guaranteed pools for non-ultra players were also nice some years ago, now I don't drink from them unless I'm on the verge of Shift+Q due to boredom/lack of significant events with a PC. I don't like the SoRS either due to the corruption, but those who do, can get what would otherwise be the best shield in the game here. Also, Darkforge provides the shortcut to the surface if you've showed the RotHK to the eternal guardian, so bonus points for that. Has anyone seen the message "the stairway is very long"? You should definitely get that when taking the shortcut, as you're going ~34 levels up/down in one hoop
3. D: 49. Hey, it's the ice level! You don't usually spend here long (not that you'd want to, if you're a Drakeling), but the interesting twist of having to move by throwing junk the opposite way is enough to make me like it. Also, you can't, or more exactly aren't meant to, bring monsters here. So, nobody/nothing to pat (or stab) you in the back - you're on your own.
I feel that it could have been done better, however. Perhaps a blueish/white color palette, being told about the ice if you 'l'ook at the floor, and making the walls undiggable would help. I might submit a feature suggestion about this soon.
Runner-up: the road tile where you start the game. I like pressing > here to equip, read and drink my starting items undisturbed.
Okay, I've written enough for now What are your favorite places in this game, and why do you like them? You don't have to list exactly three like me if you like less or more than that, of course.