So in this post I'm going to give a big giant list of all the game portals I've found that don't allow foreigners to sign up for them. You'll have to ask one of your Korean friends to make an account for you. I recommend ordering some food and having them spend 20 minutes making you an account for all of them to get it out of the way if you can. I'll be covering various games from these sites, and you might see one you like. I'm not going to detail the sign-up because I assume your Korean friend with a Korean ID can read Korean.
MGame has the requisite number of about a dozen MMORPGs, and then some other games. They have a first person shooter in the way of Operation 7, a real time tactical game that has some relation to pearl harbor, an arena combat game called Crashbattle and a couple of casual and overly cute RPGs. They are apparently responsible for Knight Online. This is a Korean MMORPG that was a really blatant Everquest ripoff in terms of interface that made its way over to North America a few years ago. It wasn't that impressive then, probably not that impressive now.
Up next, Netmarble. Netmarble again has a dozen or so MMORPGs but they have more developed categories of action, sports/fps, and casual. Netmarble has the Korean version of Perfect World. If you're familiar with that and want to play on a server with a low ping I recommend it. Since Perfect world has a few english editions it might not be a big deal to find a walk-through if you want to play it. They also have Sudden Attack which is a wildly popular but long in the tooth first person shooter. Most of their action games are fighter beat-em ups. So if you're interested in those you might want to try some of them out. There is a Gundam one. Gundam games are hard to find in the west. They have Gunz which is a game that has enjoyed some popularity outside of Korea as well. They also have a mech fighting game called Metal Rage.
Hanbiton doesn't have an overly huge abundance of games but they do have a few interesting ones. They have the only place you can still play Hellgate: London online. They acquired the IP as it was going belly up. You can download the game (huge for a Korean game over 5 GB) and play it for free. It is an action RPG so there is a lot of text, but there are walkthroughs for the English version if you can't sort it. They have a FPS in beta called Warcry that looks very interesting, and a golf game that has strangely been ported to various other game systems including the play station and playstation portable called Pang-ya. To me this site is worth it just for Warcry, it looks like it will be very good.
I'm going to mention Nexon. Technically foreigners can sign up for Nexon but you have to go to their office (and possibly still speak Korean there) or fax them a DNA sample. So if you've got a friend making accounts for you, you might as well just have them do this one up too. Nexon has Counterstrike online (with a zombie mod), Warrock (a game that also showed up in North America with an atrocious beta) and combat arms another relatively popular FPS. They also have a new cute FPS called Bubblefighter which involves water guns and forming giant bubbles around people. They have a handful of sports games and a fighter or two.
That's it for game portals that I've found that need a Korean to help you do the sign-up. I'm always looking for more and if I find some new ones I'll post them. I still have another couple of game portals that foreigners can sign up for on their own. I've also found that independent games typically don't allow foreigner sign-up on the sites. So if you see some of those you like you'll have to order some more food and get your friend back over.