Sesos, 2 bytes (non-competing)
0000000: 2c06 ,. Prints an infinite amount of ones.
Try it online! Check Debug to see the generated binary code.
How it works
The binary file above has been generated by assembling the following SASM code.
set numout ; Switch to numeric output add 1 ; Set the current cell to 1. jmp ; Jump to the corresponding exit marker. put ; Print the integer in the current cell to STDOUT. ; (implicit jnz) ; If the integer in the current cell is non-zero, ; jump to the previous instruction.