The only real limitations to "not genuine" software are that non-security updates (free programs, although I think IE is not subject to this) cannot be downloaded from Microsoft (but they would actually install if you got them elsewhere), and the nags like the 5-second wait to log in, the black screen, and the logo telling you it is counterfeit.
To download the non-security updates and free products, like Microsoft Security Essentials, you must install components of the "Windows Genuine Advantage" initiative, which validates your install, and lets you continue your download.
Even without that, the PC will continue to function indefinitely, and they can install any software packages on it they want, including legitimate software packages from Microsoft like Office. There are no other restrictions other than the download from Microsoft aspect.
The only client I ever "fired" was because of their extensive use of counterfeit software, and they had virtually nothing that was legal when I first started, and it was like pulling teeth to get them to buy anything legally.
Your client should be gently urged to buy a legitimate copy of Windows, as you have done.