How can I determine the highest GPU-supported DirectX version?
Use GPU-Z though it will only tell you for what type of DirectX it was designed, so that's often the DirectX version at the time of release.
In case you were wondering if it also supports older versions, here's a quote from Wikipedia on DirectX:
Various releases of Windows have included and supported various versions of DirectX, allowing newer versions of the operating system to continue running applications designed for earlier versions of DirectX until those versions can be gradually phased out in favor of newer APIs, drivers, and hardware.
APIs such as Direct3D and DirectSound need to interact with hardware, and they do this through a device driver. Hardware manufacturers have to write these drivers for a particular DirectX version's device driver interface (or DDI), and test each individual piece of hardware to make them DirectX compatible. Some hardware devices only have DirectX compatible drivers (in other words, one must install DirectX in order to use that hardware). Early versions of DirectX included an up-to-date library of all of the DirectX compatible drivers currently available. This practice was stopped however, in favor of the web-based Windows Update driver-update system, which allowed users to download only the drivers relevant to their hardware, rather than the entire library.
Prior to DirectX 10, DirectX runtime was designed to be backward compatible with older drivers, meaning that newer versions of the APIs were designed to interoperate with older drivers written against a previous version's DDI. The application programmer had to query the available hardware capabilities using a complex system of "cap bits" each tied to a particular hardware feature. For example, a game designed for and running on Direct3D 9 with a graphics adapter driver designed for Direct3D 6 would still work, albeit most likely with degraded functionality.
However, the Direct3D 10 runtime in Windows Vista cannot run on older hardware drivers due to the significantly updated DDI, which requires a unified feature set and abandons the use of "cap bits".
Direct3D 11 runtime will introduce Direct3D 9, 10, and 10.1 "feature levels", compatibility modes which only allow the use of hardware features defined in the specified version of Direct3D. For Direct3D 9 hardware, there will be three different feature levels, grouped by common capabilities of "low", "med" and "high-end" video cards; the runtime will directly use Direct3D 9 DDI provided in all WDDM drivers.
The best way is to just go to the manufacturer website and find your video card specs. You'll find all the information you need there. You could also try and run dxdiag from the Run menu; that will also give you a bit of information about the video card and its drivers. Look for the "DDI version" on the "Display" tab:
