How about a new gene. When it is dominant on a pet, it means that the pet gains more XP in explore only from defeating a monster, if that monster had previously defeated the pet earlier.

It could be called something like.. Headstrong or something.

And maybe, if the monster has defeated your pet
once before +20% XP (ie, your pet won on the second try),
twice before +40% XP (ie, third try),
thrice before +60% XP (ie, fourth try),
four times before +80% XP (ie, fifth try)
and
double XP if it so happens that your pet still has energy and finally defeats the monster on the sixth try or more.

I don't know how difficult it would be to code, but I think it would make the game a little bit more interesting. 

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EDIT: OR this could just be an additional feature to explore. For all pets.

Or could be a feature that requires a building to be built in order for it to apply.

You know.. I support this... it would be cool if pets had items that you could equip... It could be a tailsmen or something...

I like that it should be "headstrong" :D

I like this idea, but I feel that it could be easily be abused by users purposfully losing battles to gain more xp.

Hmm.. bluestar, you have a point... so maybe only if the battling is set to "auto"?

Out of curiosity... how does one purposely lose a battle in explore?