That really depends on the game. Take Ninja Gaiden, or Super Mario Brothers, or Castlevania 1 - the difficulty ramps steeply but your characters' abilities do not ramp at all. Zelda levels generally get harder faster than you get tougher (with some exceptions).

In some games, choosing the right advancements is a major part of the game. It's seen most clearly in Epic Battle Fantasy 2: over the course of the game, you get 10 abilities (and only 10, out of a long lineup); picking the right ones (and ensuring that you qualify for them) is a lot of the challenge of the game. There is some of a 'numbers go up' element to it, but if you don't pick the right things, you are screwed - and there's no grinding to get 'em all. The other installments in the series unfortunately lack this.

That said, I play single-player games a whole lot less than I used to, partially due to this.