Single Board Computer with no hardware memory for firmware
Almost every microprocessor (ARM, PIC, ect) on the market has some kind of ROM (usually flash) that is built in to the microprocessor. I am unaware of any SBC's  that don't have ROM. 

A processor needs a memory location to tell it where to boot from. A processor can boot from many places, such as RAM, ROM, flash (like an SD card), or even load up programs from USB, serial, or ethernet. 

Once the processor loads the memory location a program can be run to copy the rest of the program from ROM, USB, ect to the RAM. Once the program is in RAM it can be executed. 

Because of this if you want to make a device's memory less 'hackable' use a read only memory technology that can only be written to once, like a Programmable Read Only Memory (or PROM) that is One Time Programmable (which are being obsoleted fast). This means that no one could hack into your device electronically and change the ROM (but they could still change the RAM, so the exploit would go away when the device was rebooted) The disadvantage is that to change your code you would also not be able to fix any bugs (or security holes) in your program without changing the ROM chip.

A better thing to do would be to use micro controller with an encrypted rom to prevent tampering as someone would have to have the key to read the ROM's true contents.