For (quite literally) playing around at home, I'd like to get a copy of Sharepoint. I might find a use for it, so I don't want a trial version. I'm aware that there is a "free" version that doesn't include the MS Office integration etc, and that's fine. I just might use it for document storage and things.
SnapOverflow
Your after Windows Sharepoint Services, WSS for short, as opposed to MOSS which isn't free. It's available for Windows Server 2003 and 2008. Should come pre-installed with Windows Server 2003 R2, or you can download it here. Details for downloading/installing on 2008 is here.
Don't think it's possible to run it on a client OS like Windows XP or Windows 7, so if you dont have this running at home, you'll have to get something like Virtual PC or VMWare and something like an MSDN or technet licence.
Youcan install WSS 3.0 on a Windows 7 box. It does require you to modify the local machine gpo. You can find information regarding that Here. And as far as sql goes you can use sql server express.