1. Increase running speed.
2. Increase jumping and safe fall disance.
3. Increase swimming speed.
4. Increase the amount of time you can hold your breath under water. There could be all kinds of underwater content this aids with.
5. Establish and develop an Out-Of-Body ability; the more trained you are, the longer you can stay out-of-body, scouting locations or just wandering around.
6. Raise base stats.
7. Learn new dances.
8. Learn how to play a musical instrument and then learn different tunes. If anyone ever played Asheron's Call 2, that game had the most advanced interactive musical instrument system I've ever seen in a game. Being able to develop musical skills would be great, especially if very advanced musicans could actually buff characters in a meaningful way or open the door to certain storylines/quests.
9. Gymnastics, opening the door to all kinds of cool animations and moves, even in battle.
10. Crafting, of course, which can be any number of paths in itself.
11. Weapon specializations that increase one's effectiveness with very specific weapons.
12. Swordplay - opens up new combat animations with various weapons which the player can substitue for standard animations.
13. Spellcraft - same as swordplay, only with magic-based animations.
15. Ability to use any specific armor, magic, or weapons.
16. Job - ability to apply various other skills and abilities towards a job or career that earns the player money while offline. For example, if one develops their gymnast ability, they can become a certain level of a gymnast teacher through this offline progression avenue (but only to their corresponding level of gymnastics training). They can then, instead of advancing in some skill while offline, set their character to be at their job and earn money while you're offline.
17. Bartering - buy items in shops for less, get more money for items you sell to NPCs.
18. Increasing various factions via oflline service to those groups.
19. Learning new emotes.
20. Cartography - grants detailed maps of various areas of the game (especially if the game has a "fog of exploration" style reveal.