Don't spend anything (money, debit card, credit card) for an entire day. Sounds simple but you will need to do a little pre-planning. You will need to have enough gas in the car to get to where you are going, bring your own lunch/coffee/water, not have any last minutes bills to pay, and say "no" to the kids when they ask for cash.
Don't spend anything for an entire week. Obviously this will require even more planning. You will need to fill up your gas tank before the challenge week then limit your driving to make it last for the entire week. You will need enough food in the house to feed everyone for an entire week. You will have to make sure the bills that are due during the week will have been paid.
Finally, for the hard-core challengers, could you go an entire month without spending anything? You would need to pay your bills for the entire month, buy some prepaid gas cards to use throughout the month (and make them last so you don't end up with more month than gas), you would have to stock your home with a month's worth of food and supplement it with produce from your garden. You would have to be strong when mid-month someone (the kids, the spouse, you) are having a consumer meltdown. You would have to get creative--bartering, trading, foraging for food, fishing, basically come up with a multitude of ways to meet you and your family's needs for an entire month without spending any money at all.