Just to elaborate: Lew's correct. You can own an unlimited number. I know investors who own 20, 30, or more investment properties.
And Robin's link is a good one: With conforming loans, under certain conditions, you can own up to 10.
What that comes down to is how you own the homes. Lew's not quite 100% accurate. There can be mortgages on more than 10. It's just that your name may not be on the conforming ones. And that opens two possibilities: There are some noncomforming mortgages out there. The second option is to look for properties with owner financing. Or purchase "subject to"--meaning the existing loan stays in place. You're deeded the house, but the mortgage stays in the name of the seller.
Purchasing "subject to" or with owner financing (or doing a lease-option) accomplishes something else: It conserves your funds. You'll run through a lot of money very quickly if you have to put 25% down on every investment property. But if you can acquire or control those properties with little or no down payment (as with a subject to, owner financing, or a lease option), then you can buy more properties and not end up cash-poor.
Hope that helps.