I managed to finish all of these quests in my game before I started fighting (after the tutorial fights, that is)...
traveler's diary 4590xp
starving dersu-kumata 3740xp
bagouri's love 3400xp
pirate treasure 3400xp
amulet of illumination 170xp
king's bride 935xp
swill antidote 4590xo
a message from shionis 7480xp
message from the dead 6800xp
demon in stone 6120xp
bottled conscience 382xp
contraband searcher 425xp
trading license 76xp
My game hadn't spawned a Tekron map except on Montero, so I was kind of stuck at that. If you also get a Tekron map, add the following...
submarine 3740xp
secret of the superlative soap 3060xp
treasures of the abyss 3060xp
glasses for old bobur 3060xp
I accidentally missed...
secrets of the temple 8500xp
mercenaries for sohaty 382xp
For a total of 63,910 experience available from quests without fighting at all.
I deliberately skipped...
Plight of the Slave, for 6800xp, because it requires you take a less-rewarding route than fighting Asklepy.
K'Sar's memory, for 8500xp, because it locks you out of the cave before you kill the hero, who is one of the most rewarding heroes in the game.
If you don't mind doing just one fight, to clear the bridge to Monteron (which can quite reasonably be done with just one troll if you're a careful mage), add the following quests as well...
falling star Je-Lo-Pi 3400xp
traveling for beer 765xp
the terrible study of necromancy 1870xp
It might be possible to do Treasure of Jack Albatross (for 6120xp), but I suspect for some bizarre reason The Statue of the Dark Warrior is a prerequisite quest (I can't imagine why I didn't get it earlier otherwise).
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