Tad,
Yes it's possible. I think Ben is right, that it could be done cheaper and faster in another aircraft with a transition to the Extra. Also, insurance would be a problem. I don't know of any flight schools that have student solo insurance for an Extra 300 or any companies that will insure an Extra or a Pitts to a student pilot. This is usually the deal killer in my experience.
If you own the plane already or are planning on buying an Extra, then it might be more possible, but lack of insurance would expose you to some pretty severe liability should you taxi into a Cirrus or Bonanza or even some car. You would also have to really look around to find an instructor willing to train someone to fly the thing without insurance. The chances of that instructor being any good are slim, because instructors that take those kind of chances usually aren't all there.
With these things in mind, it's totally possible to get some training in an Extra 300 before finishing your private license. Whether it's aerobatic instruction or unusual attitudes or even some pattern flying, with a good instructor you could actually learn something and enjoy flying the Extra right away. It may not be the cheapest route, but it might be fun without adding excessively to your training cost and training time.
Chris