Can you think of a specie of animal that, other than to reproduce, has no other purpose?

Yes.

Well, in the end it's all pointless so I'd say nothing really has a purpose. I still love critters though, my piggly-wigglies are so cute even if they nibble a little too hard sometimes..

All animals and insects have a purpose.  We may not know what it is.

Ok, maybe not the human animal.  Unless its purpose is to destroy its environment, every other species, and itself.  It's REALLY good at doing that.

The Lake flies that emerge from Lake Winnebago in the USA and also in Lake Victoria in Africa......There are countless trillions emerging from all lakes ,They appear from the to mate and then die.....their eggs get deposited back into the Lake and the whole process starts over again....They are born to die and to be eaten by many bigger  predators.....
The African people that live around the Lakes catch and eat them as an health sorce of protein :) 

They only eat fly'ed food as they ain't got any ovens..... :( D 

You probably already knew this Einstein but see if this is the answer you are looking for:
https://mentalfloss.com/article/57800/horrors-anglerfish-mating.

I believe the frogs who hibernate during the dry season only come out to reproduce during the wet season.