Remember that your computer is really made of multiple computers that run bare-metal code or can have their own OSes. For example Intel CPUs have Minix running inside of them.
>> After the memory controller, bus bridges, GPUs, and DMA controller, it's probably the largest one
Probably best to distinguish between processing-heavy and throughput-heavy; apart from the GPU, all those are throughput-heavy but do very little to the data, and I think would not normally have a recognisable processor or an OS. There's the additional problem that they're required to boot, so nowhere convenient to load the OS from unless you scatter SPI flash across the board.
GPUs on the other hand have a full task-switching operating system.
It means it has a large processor that can run an RTOS. It does not automatically mean it has an RTOS.
I would be surprised if the faster bus and memory controllers had it (because of latency problems). I would expect something like it on a USB controller, GPU, network or disk interface. I really have no idea what to expect from a DMA controller.