Since they usually start small, they'll just either start off at a free host, or a cheap virtual host (like http://www.below10host.com
) - then move on to maybe using a reseller program to get space at cheaper costs (and there are plenty of examples of such hosts).
Once they get pretty big it's time for them to make the jump to co-lo/dedicated servers. Usually you'll find that the majority of the 'huge gaming sites' are placed on dedicated servers, such is the amount of traffic they consume.
Just remember : there ain't anything such as unlimited.