BB(8)
The 8-state, 2-symbol busy beaver problem BB(8) is unsolved and largely unexplored. The current champion is the BB(7) champion 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF
(bbch) by default. It is thought to be almost certainly larger than Graham's number.