One of you biggest investments when inspecting taking on an adventure race is your mountain bike. Generally, the boat for the paddle section will whether be provided for you by the race organizers or you will rent it. The gear for the trekking and navigating sections is minimal, and many races do not even contain a climbing section anymore, so save the bulk of your budget for your bike.
With that said, you still don't want to break the bank. Here in Colorado it's not unusual to see a car valued in the hundreds of dollars hauling on its roof a bike worth thousands. Sure you can get a mountain bike with full suspension, the absolute lightest components and all the bells and whistles but remember, puny Timmy is going to need braces someday.
By the way, unless you are taking on an urban assault, it is pretty much guaranteed that portions of the bike section will be on dirt roads, trails, rocks and other surfaces which will set in motion a road bike into a formal protest, so let's look at the two types of mountain bikes you will want to consider, hard tails and full suspension bikes.
