So I got this idea from the quiz site, Sporcle. In it, players had to click on an equine, feline, canine, feline. 
For example: zebra, tiger, wolf, persian until the options were used up.

I decided to bring it here, but altered for forum use.

I will name an equine. Player 2 will name a feline. 3 will name a canine and 4 will name a feline. And so on.

Acceptable: breeds, species, mythical (not one you made yourself).

SHIRE

Maine Coon! 

Java pony

Russian blue

doberman shepherd

Ragdoll

Pegasus

lion

newfoundland

liger

knabstrupper

ocelot

dalmatian

Savannah

caspian

Manx

Shiloh Shepherd

siamese

Tiger Horse! 

snow leopard

Doberman