I'm not sure if it's to do with game design, or rather the way I personally approach playing games, but I tend to enjoy the final levels of most games way more than the opening ones. Certainly this is true for me playing N64 games especially, might not hold so true for older 2D games.
Whenever I start a new game, apart from rare cases it usually takes me a good couple of hours to really find my groove with it. In fact, there are many many games I own where I've played the first 2 to 3 hours and then put them aside, knowing I want to play more but just not being ready to really get into them and enjoy them yet.
However, by the time I reach some of the later levels I am fully on board and totally clicked with the game, so I enjoy them way more. A significant example of this for me is Armorines, which for most of the game I found totally average, but the final levels in the alien mothership were freaking fantastic. Also, Turok 2 just got better and better imo. I was almost ready to give up on level 2, but every level after that just got me more and more into it, until the final level which I consider a masterpiece.
Then there are whole new genres to consider, for example puzzle games. No one can say with a straight face that the early levels of puzzle games are the best, it's when they get DEVIOUS that they are the most satisfying.
Then again, as I said above this might not hold true for some games, especially NES, SNES and Megadrive. We all know that chemical plant zone is the best level in Sonic 2, for example, and that's only level 2!