Can crackleburr respond to himself?
A player could, in fact, repeatedly activate Crackeburr if the player had the mana to do so, as long as the player had two other creatures available that were both red and blue, because the tapping and untapping are costs and not effects.

Each time the player would have to pick targets for the effects - and could keep picking the same targets over and over again if desired. The effects with targets would go on the stack, so that you would have a stack consisting of the three-damage effect at the bottom, then the return-to-hand, then three-damage, then return-to-hand, etc. Instances of either effect would fizzle if they targeted creatures that were already returned to hand or destroyed via damage. (Bear in mind that damage doesn't use the stack, so as soon as a Crackleburr invocation is resolved because the stack unwound to that point, then if the creature has lethal damage, it will be destroyed.)

Presumably your opponent used Crackleburr's untap ability specifically to bounce itself. This would work, and yes, your opponent could tap it a third time to do three damage if there were mana and untapped creatures available. At that point these things happen in order as the stack unwinds:


Crackleburr does three damage to the second target
Crackleburr returns itself to its owner's hand
Path to Exile tries to exile Crackleburr but fails because the target is no longer on the battlefield
Crackleburr does three damage to the first target