bit bang programming with a 1/222.2ns clock
"Bit banging at 222.2 ns" presumably means transferring data via a say 8 bit port or one pin on a port with a cycle time of 222.2 nS.

The 222.2 figures suggests a degree of accuracy which would require a timer or isochronous programming to achieve, and if you want this precise to the number of figures specified you will probably have to select your system clock to suit - probably by use of a PLL.

The simplest program involved is liable to be of the form


Get data via index pointer  
Adjust index pointer  
Write data to port   
Test for end of data  
Go around power please (aka loop)


If you can achieve each of these in a single processor clock cycle (only in some systems) then you need each instruction to take 222.2 ns / 5 =~ 45 nS.

Which would require a 22 million instructions per second processing rate.
Realistically one or more of the above step is going to take more than one clock cycle, and you'd like more sophistication than the above, and you'd probably like time to breathe as well. So a processor with a clock rate in the 30 MHz + range that can handle most instructions in 1 clock cycle and ideally a 50 MHz plus part would be better.

If you are trying to eg implement an asynchronous UART (unlikely at 4.5 Mbps) then youwill need a "substantially more capable" processor.

A small amount of support hardware (internal or external) may ease this task somewhat.

A Cortex M3 should do this "very nicely" if clocked adequately.
 You can buy ST's bottom end version of this in 1's for about $US2.20.
 STM32F101 ... ARM Cortex M3. It's SPI is capable of 18 Mb/s and may well constitute the "small amount of support hardware" mentioned above.