I'm not sure why you would think that changing a register would require kernel intervention. Some special registers may be privileged (those controlling things like descriptor tables or protection levels, with which user-mode code could bypass system-mode protections) but general purpose registers can be changed freely without a kernel getting involved.
When your code is running, the vast majority of instructions would be things like:
inc %eax
movl $7,%ebx
addl %eax,%ebx
As an aside, I'm just imagining how slow my code would run if it required a system call to the kernel every time I incremented a counter or called a function :-)
The only thing I can think of would be if you thought your execution thread wasn't allowed to change registers arbitrarily since that may affect those registers for other threads. But the kernel would take care of that when switching threads - all your registers would be packed away somewhere for later and the ones for the next thread would be loaded in.
Based on your comments, you seem to think that the time of adding is when the CPU protection mechanism should step in. In fact, it can't at that point because it has no idea what you're going to use the register for. You may just be using it as a counter.
However, if you do use it as an address to access memory, and that memory is invalid somehow (outside of your address space, or swapped to disk), the kernel will step in at that point to rectify the situation (toss your application out on its ear, or bring in the swapped-out memory).
However, even that is not a privileged instruction, it's just the CPU handling page faults.
A privileged instruction is something that you're not allowed to do at all, like change the interrupt descriptor table location registers or deactivate interrupts.