Does azure support things like mongodb and redis?

You can very easily run mongodb in Windows Azure. I presented this at MongoSV - video here.

EDIT: In December 2011, 10gen published their official MongoDB+Azure code on github. This contains a project for replica-sets, as well as a demo ASP.NET MVC application (taken from the Windows Azure Platform Training Kit) that uses a replica set for its storage.

Standalone servers are straightforward, except you have to deal with scale-out: you can't have multiple instances of a standalone server simultaneously, so you'll need to plan for this: take all but one out of the load balancer, or only launch mongod if you can acquire the Cloud Drive lock.

Replicasets are doable, as I demonstrated at MongoSV. However, I didn't cover the intricacies of graceful shutdown of a replicaset to ensure zero data loss.

You can run memcached as well - see David Aiken's post about this. Note: Now that the AppFabric Cache service is live, you should look into the pros/cons of using that over memcached. Cost-wise, AppFabric Cache should run much less, as you don't have to pay for role instances to host your cache. More info about AppFabric Cache here.

You now also have the option of running Redis in Windows Azure on Linux virtual machines ! In the case of Redis, this would allow you to use the "official" build instead of the "unsupported" Windows build ... For MongoDB, both choices seem equally valid (running on Linux virtual machines, "plain" Windows virtual machines, or using 10gen's package to run on "managed" VMs (Cloud Services).

FYI, there's now a Redis installer for Windows Azure available from MS Open Tech (my team). Here's a tutorial on how to use it: http://ossonazure.interoperabilitybridges.com/articles/how-to-deploy-redis-to-windows-azure-using-the-command-line-tool