Ever notice how certain games manage to get more and more fun the longer you play? (not indefinitely of course) For example, Psychonauts started off pretty meh, making me wonder what all the fuss was about, but luckily I persevered and ended up having a blast. (Special mention : milkman conspiracy). It's not a perfect example because the game had some low points for me, specifically the the bull run level and the meat circus. Not that it matters. Both are completely overshadowed by the awesomeness of the Milkman Conspiracy level!

Not that this only applies to games that start of pretty 'meh' either: case and point, Jagged Alliance 2, which definitely isn't 'meh' in the beginning, only a bit frustrating. At least with the 1.13 mod, part of what makes the game more and more fun is the clever balancing. On the one hand the enemy becomes more elite, numerous and tenacious. But on the other your guns become bigger and badder, and your mercs more professional. This is balanced in such a way that the fights become shorter and more intense as the game progresses, while never losing the element of real and serious danger. Even in the late game, no matter how well you are armed and armoured, you can still get slaughtered in the blink of an eye if you're not careful. This it seems is a perfect recipe for gameplay that becomes more fun the longer you play. Towards mid game you start to feel like you command a group of colorful characters worthy of a direct-to-video action movie set in some third world country, complete with over the top action, gratuitous violence and relentless pyrotechnics!

Morrowind and Oblivion are two more examples, though unfortunately there the gradual build up of 'fun' peaks long before I'm anywhere near to completing the game, causing me to lose interest. I've started and stopped morrowind and oblivion about 10 times by now.

Resident Evil 4/6.

The first 3 games I experienced that was with Pirates ( Sid's...the original game), Railroad Tycoon ( again the original, from the same stable) and Falcon 3.0 (Spectrum Holobyte) . And I mean, when these games were released. I can probably Yes that's 25 years ago. Then there was Aces of the Pacific and quite then other games. Then I left gaming for a while, and when I came back to it, it became much harder to get that hooked to a game. Recent games that had that effect on me : Wastelands 2, Alea Jacta Est ( well, this one has a hell of a learning curve), Hegemony (Yes, that game GOG refuses ) and Crusader Kings 2.

Stalker has a pretty rough beginning, I think, but once you get up and running it becomes amazing.

It might be hard to appreciate how great Spelunky is from early games, but once you put more time into it you start realizing what a marvel of game design it is. Lots of simple game elements that can bounce off each other in complex ways, and when combined with the randomized level creation you get a game that is always stimulating to play.

Morrowind

Terraria

I got that with the Stalker games. Starts off kinda slow as you're learning the mechanics of the game, how to deal with the combat, etc. By the time you get those down to second-nature, you're starting to get into the interesting lore.

Way back when, Railroad Tycoon also felt that way for me.

There's a long list of games that improved with increased exposure, but I'll go with those two since they first popped into mind.

Afterfall Insanity: Extended Edition.

The first half or so of the game is rather ho hum survival horror action-y stuff. Not bad, just not particularly interesting either. The second half is well...it doesn't become a comedy, but it kinda stops trying to be serious, if that makes sense? Plus the levels become a lot less generic the further you go, especially once you hit around the halfway point.

One minute you're fighting your standard demonized security guards with your standard pipes, fire axes and rifles in a vault while trying to save your girlfriend, the next you're eating an arm and fending off cannibals using a sawblade taped to a stick and discovering the reality of what amounts to a boutique store window. It really goes off the rails in a great way, and what makes it so satisfying is that the main character has similar reactions to yours.

Magic Carpet

That's standard procedure in any TES game! ;-)

Oh yes how could I forget about stalker? Especially when you hit the underground labs for the first time, and the really freaky supernatural things start to mess with you :P At least when it comes to the original. Not too long ago I played clear sky for the first time, and was disappointed to see that even though most of it takes place in the exact same locations as the original, you never get to visit the underground labs again.

Another one I can think of is The Sims, but in a different way. I very quickly got bored with it the first time I played it, and proceeded to shelve it. However, a week or two later I found myself with this weird urge to give it a second try, and then I loved it. Really weird... It's almost as if the game planted a seed in my brain or something :P

[PROTOTYPE]
TMNT

Spec Ops: The Line was one of those experiences for me but in an odd kind of way. It started with pretty enjoyable gameplay and a meh story, but at a crossing point the gameplay turned meh and the story bloody amazing! So I began playing for the gameplay and finished for the story.

Gameplay wise I had trouble enjoying Dark Souls 2 the first time I tried it (like I guess most had) but it really grew on me with the lore and world design, I hold Majula as one of the most beautiful virtual places I've ever had the pleasure to visit.

Funny, that is exactly how I felt by the end haha. The story took a dark turn. Loved it.

Magic Carpet was so cool! I also remember the sequel: Magic Carpet 2. :)

Final Fantasy X. I just wish it was longer, as real game started in Calm Lands...