Almost all major no-load mutual fund families allow you to do the kind of thing you are talking about, however you may need an initial investment of between $1000 to $3000 depending on the fund. Once you have it however, annual fee's are usually very little, and the fees to buy that companies funds are usually zero if it's a no-load company (Vanguard, TRowPrice, etc) With the larger companies that means you have a pretty large selection of funds, but generally EACH fund has a minimum initial purchase, once that's met then you can buy additional amounts in small quantities without a problem.
For someone on a smaller budget, many low cost brokers (ETrade as mentioned by Litteadv, Scottrade as mentioned by myself in another similar question today) allow you to start with smaller initial balances and have a small selection of funds or ETF's that you can trade from without commission. In the case of Scottrade, they have like 15 ETF's that you can trade comission free. Check with the various low cost brokerages such as ETrade, Scottrade, and TDAmeritrade, to see what their policies are, and what if any funds/ETF's they allow you to trade in without commissions.
Keep in mind that for Mutual funds, there may still be a fund minimum initial investment that applies, be sure to check if that is the case or not. The lack of any minimum investment makes ETF's a slightly more attractive option for someone who doesn't have the 'buy in' that many funds require.