Lets say I want to do a bit of experimenting with some high bandwidth uses for transmitting data, high quality voice, etc. Which Amateur band will give me the maximum bandwidth, and what is the maximum width?
Without a doubt, the 300GHz band. All frequencies above 300GHz are authorized for amateur use. There is no bigger band, amateur or otherwise.
Here's all the national band plan has to say about it:
All modes and licensees (except Novices) are authorized Amateur Bands above 10.5 GHz.
Also, it has the coolest band designation: Tremendously high frequency.
While there may be some restrictions on what you can do here, they are woefully outdated and likely to be revised. Given that you stick to "good amateur practice" (essentially, play nice), I really doubt you will be arrested. The 300GHz+ band is all about experimenting with weird stuff, so it would almost be bad practice (and thus illegal) to not try some extremely high-bandwdith experiment there.
If you intend to follow the national band plan, your best options lie within the 3300MHz to 3500MHz band, specifically at 3360.000 to 3400.000 and 3460.000 to 3500.000 you are allowed up to 22MHz bandwidth transmissions.
Your next options occur in the 33cm and 70cm bands:
420.000MHz - 426.000MHz - ...and experimental 909.000MHz - 915.000MHz - Analog and Digital broadband multimedia 915.000MHz - 921.000MHz - Analog and Digital broadband multimedia 921.000MHz - 927.000MHz - Analog and Digital broadband multimedia
Check into the bandplan further, and look for local and regional bandplans to make sure you aren't stomping on anyone else's signal. Otherwise keep in mind that one of the primary missions of Amateur Radio is to experiment and advance the art of radio communication. It is likely that you can use more bandwidth in some areas outside the bandplan if you coordinate with others, and the bandplan can be changed if your use provides new opportunities to everyone.