Nope that's not what happens. In fact that is prohibited in the bill.
Yep, the bill specifies that taxes will be collected on taxes. 23% of gross payments, and gross payments includes the taxes, so it's 23% of the purchase price + 23% of the taxes, and then 23% of the taxes on the taxes, and then 23% of the taxes on the taxes on the taxes, ... until it all adds up to 29.87% on the item itself.
Thanks for clarifying that the FairTax taxes the tax multiple times.