Depends on your definition of credit card, and active. 5 general purpose, plus cards for Home Depot and NewEgg.
The point of having more is 1) They give you free money when you sign up; and 2) a year or more at 0% interest. A couple of years ago I had only one general card, then MMM pointed out the free money aspect. I don't overdo it, but adding up the signup bonuses, cash-back, and profits from keeping free money in the market for a year, I've made several thousand bucks for very little work.
PS: Of the 5, only 3 are really active. The other two have 0% balances that are being paid off at a bit over minimum until the 0% runs out.
For a list:
1) Chase Freedom. Oldest card, has a few automatic payments (phone, internet, etc), otherwise only gets used when the quarterly 5% cash back category fits something I buy (like gas this quarter).
2) B of A Visa. $100 signup, gets used for 2-3% back on gas &c.
3) CitiBank ThankYou Preferred. Inactive with a 0% balance, mostly because their 1% cash-back points work out to around 0.75% in practice.
4) Chase Slate. Another inactive. They offered no-fee balance transfers and 0% to the end of 2014, so I moved the balances from other cards.
5) CapitalOne Quicksilver. Just opened last month, $100 signup, 1.5% cash back, and 0% to the end of the year.