A Trophemon is a very rare bird. There are only 4 known living Trophemons. They look like eagles with antlers. If you play the game long enough, and visit the park, there will be a group of people saving the Trophemons from extinction. They will also mention a millionare trying to steal the species for his collection. Depending on your choice, you can choose the activists or the eccentric billionaire. Either way you choose you will still get a Hoverboard when you find all four Trophemons. The activists give good karma and charm, the millionare money and negative karma. (Note that you must give them all to one group. If you split them up you can't get the hoverboard)
How to Obtain
1) Get the most expensive burger from McSticks, the Grand Slam Burger, for $75. Give it to the construction worker at the end of the second island. Recieve the ladder. Help a nearby person by rescuing her cat. Receive the first Trophemon.
2) Get a Pizza and a Milkshake from Joe's and McSticks respectively. Give them to the DJ at the Night Club during the evening.
3) Get some smokes, a Cuban Cigar and a lighter. Give them to Vinnie's Goon at the bar next to Steel and Hawks.
4) There's a person on the second island playing a tune about Stick RPG. Continuously throw $25 in his hat and receive the final Trophemon. Note that you can't donate him any more money after this.