I have hired a new babysitter, who is concerned about having to file income taxes. What would be the maximum income she could make in 2010 without having to file taxes?

Assuming this baby sitter is a dependent of her parents, she has to file federal income tax if her earned income is over $5700. This assumes that her unearned income (from bank accounts or other investments) is less than $300.

But you should know that if you pay her more than $1700 for the year that you are responsible for withholding income tax and FICA. Here's an article that explains it:
The Nanny Tax at SmartMoney.com
And from the IRS:
Publication 926 (2010), Household Employer's Tax Guide