That's right, some real numbers can be easily defined while being arbitrarily difficult to calculate the game result with. But there is another reason why we want to tighten the restriction for a submission beyond the standard of being able to "figure out whether it's the closest easily".
The point of the game is for people to try to submit 2/3 the average guess. In order to calculate 2/3 the average guess, you need two operations: addition and division with nonzero divisors. The rational numbers form a dense set (for all a<b there exists c such that a<c<b) that is closed under these two operations. It is the natural playing field for this game.
The real numbers are constructed in a way that is unrelated to the structure of this game. (I think one typically invokes the concept of sets of rational numbers to construct the reals.)
If you want to see why allowing all real numbers adds nothing to the game play, just note that every real number is equal to a terminating decimal plus a real number smaller than Epsilon, where epsilon is made to be much smaller than the difference between any two submitters' numbers.
This is the only type of scenario when submitting an irrational number could help you:
You are playing against the submissions π, 2π, 7π. If you submit 2π, you tie for the win, while if you submit a rational number very close to 2π then the 2π submitter wins. The scenario can happen with any irrational number, not just π. The irrational numbers just serve to add pointless additional elements to our already well-structured rationals.
I hope I have clarified my previous comment.
I'd like to play a game with you. Send me, privately, a real number between 0 and 100, inclusive. (No funny business. If you say "my age", I'm going to throw it out.) The winner of this game is the person who, after a week, guesses the number closest to 2/3 of the average guess. I will reveal the average guess, and will confirm the winner's claims to have won, but I will reveal no specific guesses.
Suppose that you're a rational person. You also know that everyone else who plays this game is rational, you know that they know that, you know that they know that, and so on. Therefore, you conclude that the best guess is P. Since P is the rational guess to make, everyone will guess P, and so the best guess to make is P*2/3. This gives an equation that we can solve to get P = 0.
I propose that this game be used as a sort of test to see how well Aumann's agreement theorem applies to a group of people. The key assumption the theorem makes--which, as taw points out, is often overlooked--is that the group members are all rational and honest and also have common knowledge of this. This same assumption implies that the average guess will be 0. The farther from the truth this assumption is, the farther the average guess is going to be from 0, and the farther Aumann's agreement theorem is from applying to the group.
Update (June 20): The game is finished; sorry for the delay in getting the results. The average guess was about 13.235418197890148 (a number which probably contains as much entropy as its length), meaning that the winning guess is the one closest to 8.823612131926765. This number appears to be significantly below the number typical for groups of ordinary people, but not dramatically so. 63% of guesses were too low, indicating that people were overall slightly optimistic about the outcome (if you interpret lower as better). Anyway, I will notify the winner ahora mismo.