Hi Spaghetti,
Have not seen this question before, sounds interesting - perhaps someone here may have some advice from experience. You may want to provide some additional details on your plans. For example, what type of video games to have in mind, something original you have developed or simply links to other sites - one has to consider any legal challenges that may be associated.
Square is great for small businesses starting out as they do offer a free website plan but, it does have limited customization options. The paid plans do offer more features and options. Depending on your skill set, you may want to check out the Developer Forum - it's a bit more advanced when it comes to custom work.
