Well, you can go back in time, do whatever you feel like doing (beat the crap out of your boss, for instance), then return to the present with no reprecussions. Or you can "re-do" a failed mission to practice different methods of performing it (you'll have to kill your past selves, of course, but none of it really matters).
Gambling. Particularly fast-moving games like roulette. "I put it all on red." "Oh! It came up Black 14!" *Ultimate Chronological Manipulation!* "I put it all on Black 14."
Heck, the once-per-week thing actually makes this more fun: Imagine some random stranger waltzing into your casino, placing $50,000 on Black 14, winning, then immediately cashing his chips in and walking out nonchalantly.
No, only things you affect are rerolled, like the power Time Regression (which this ability basically is, except more powerful and with no XP cost).