What is maximum number of bits a CPU can process at once?

That will depend on both the architecture and implementation of the CPU.

The maximum number of bits that a CPU may process at once usually depends on its "register" size, but there are many other variables that influence and change this limit for a specific CPU. For example, a 64 bit processor may operate on 64 bits at once. Some processors may also have subprocessors that can handle even more bits at once, but those are usually not included in the processor's "bit size". As an example, Intel processors have MMX instructions that can handle up to 64 bits simultaneously, although the primary CPU is 32 bit. The largest known processor at the time of this answer operates on 128 bits at once, and there are rumors of even larger bit-processors on the horizon, although their power is largely unnecessary at this time. Some processors also have multiple parallel function units that can be running at the same time (e.g. integer units, floating point units, load/store unit) each of which processes a Word of the appropriate number of bits for its data type simultaneously with all the other function units, this dramatically increases the possible number of bits processed at one time with no increase in "register" size. There have also been CPU architectures with large Word sizes but some implementations of those architectures processed the bits in smaller groups to save cost (e.g. IBM System 360/30 like all System 360s had a 32 bit register size but processed it only 8 bits at once). Many early computers used serial ALU implementations, one that I know of had a 48 bit "register" size (stored in a continuously recirculating memory) but processed that only 1 bit at a time in the serial ALU (2 bits at a time when executing its square root instruction, but it still had to wait 2 bit times to get those 2 bits from the recirculating memory, so there was no speed increase).