Probability doesn't work like that. Likelihood has an upper bound at 1, and log likelihood has an upper bound at 0. Likelihood just represents the probability you assigned to the actual outcome. You can't have a probability higher than 1.
You can have infinitely high scores if you use log odds though: http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/
But that can only happen if you get every question 100% right. Whereas getting a single question 100% wrong is easy to do.
Whereas logarithms make multiplication just addition, and they get small very very slowly.
Personally I prefer geometric mean. Which is equivalent to the average log likelihood. It has an upper bound at 1 and lower bound at 0, and represents the "average" probability you assigned the correct answer.
A potential answer is: this asymmetry is similar to incentives outside the classroom. Claiming 100% certainty and being wrong can be disastrous to your reputation, and likely much more negative than the positive benefits of claiming 100% certainty and being correct.
You could make an arbitrary scoring metric that does whatever you want. But it wouldn't be principled, or have nice mathematical properties like this.