For the past little while I've been helping to build the just announced service called MondoCloud. In a nutshell, it allows developers to do easy server-side game logic as well as real-time messaging services.
This is a great solution for features such as in-game chat, lobby systems, and most multiplayer needs. The only multiplayer requirements it really wouldn't be suitable for are tracking positions in real-time. For example a multiplayer FPS game would not work well. But an RTS game could work if the mouse-click actions are sent to the other players.
The best part about this service is that MondoCloud provides libraries for Flash and Unity so you can easily integrate it into your game. You can see an example of a chat system on the MondoCloud website. It demonstrates the MondoCloud messaging service in Flash, Unity, and HTML5.
Combined with dimeRocker, you really have most of your online web game needs covered. And it's extremely cheap compared to hiring an entire web team and running your own servers.