Opinion on that varies, but IMO, assuming you've already "paid your dues" with papertrading, and you trade relatively infrequently so you don't get eaten up by commissions, you can start with as little as $5k. Just don't expect it (or even try) to quickly become $10k.
I use $5k as a minimum because I feel you should never commit more than 20% of your portfolio to any one position, and $1k worth of any stock is about the least amount you want to own if you want to make a profit after commissions.
In a larger portfolio ($100k), I feel that 10% per position should be the max.
People who have million-dollar portfolios tend to be conservative about the size and riskiness of their trades. That's how they got million-dollar portfolios.
Edit: One of my favorite things about one of SI's most colorful characters, Anthony@Pacific, is that he has always preached asset allocation management. Put all of your money into just one stock and he'd come down hard on you for that.