You can't really avoid them, even in some non-rpg games, but they are many times a necessary evil in many games. (along with other things, such as random battles).
I have many projects in progress, and often have many types of quests throughout. While many are oriented towards the basic style of the main game, often you can find other types of quests.
I enjoys some puzzles, and (destoy and) gather quests myself, but I often have to do many other quests in between.
My newest project has many jobs (classes) you can switch between, and each have their share of quests you need to do to increase your standing, and your level, with some odd bonus and penalties:
>A hunter gets less EXP for certain kills, but more drops of certain items from beasts.
>An alchemist gets no EXP from battle, but EXP for completing jobs making items.
>Soldier types get more EXP for battle.
What quests do you enjoy?
What makes you feel like you accomplished something?
Did I miss any quest types?