Perhaps the gamemakers thought that after the one drinking contest in his/her life, the Dragonborn will never want another one

On topic - not necessarily games strictly speaking, but here goes:
- how about brawls, like those you already have at certain points, only as random encounters; you could win money or a small valuable
- conversations with alchemist and mage characters where you would have to persuade them to teach you a potion recipe, some enchantment or spell
- a Skyrim equivalent of some dice game, for cash or valuables
- using services of a travelling fortune teller to get to know your future (they could use, for example, small stones with runes on them)
- knife-throwing contest? as a mini-game where you have to aim
- for mage characters - an animation where using spells, you do something like a mini indoor firework display
- or acting as a bard and doing something guitar-hero-like; depending on how well or bad you do people start booing your or dancing (ok, I admit... this one would be a MAJOR copyrights issue)