I need to install SQL Server 2005 Express on Windows 8 Home Edition x64.
But I get a compatibility error!
I have tried SP1, SP2, SP3 of SQL Server 2005.
Visual Studio 2008 is working fine.
According to this Microsoft KB article, SQL Server 2005 (in any form, even with all the Service Packs installed) is not supported on Windows 8.
SQL Server 2008 is supported, as long as you install Service Pack 3.
SQL Server 2008 R2 is supported, as long as you install Service Pack 1.
SQL Server 2012 is supported with any version.
While the others are correct in that its not supported, it doesn't mean you can't install it and that it doesn't work (actually I have SQL Server 2005 on my Win8 machines, and have not seen any issues with it) If you turn off the Compatibility Assistant (which actually will block you from doing what you want) you should be able to install it. See this answer at Superuser which describes how to turn it off.
As an alternate suggestion, maybe you could setup Win7 in a hyper-v virtual machine, assuming you have a license for Win7. If you don't you may want to consider at least an MSDN OS only subscription, which would cover you in this situation (you could install Win7 or whatever OS you wanted that SQL Server 2005 was supported on as a virtual).