0 1 2 3
A 1RB 3LB 1RZ 2RA
B 2LC 3RB 1LC 2RA
C 3RB 1LB 3LC 2RC
with the triple (t',d, s') in row s column t specifying the transition from state s with symbol t under the tape head. the machine overwrites symbol t with t', moves the tape Left/Right according to direction d, and changes state from s to s', halting if s'==Z.This is 3*4*log2(4*2*log2(4+1)) or about 64 bits of information. Meanwhile, in only 49 bits, BBλ(49) far exceeds Graham's number [1].
If you run it for a bit you see what's going on, State B turns 0's into 2's and 1's into 1's transitioning to C and state C turns 3 -> 2's and transitions to A. So you just iteratively lengthen your run of 3's exponentially since it requires a full pass through all the 3's to fix a 2->1.
Here's a tangent to explore:
A BitGrid[1] (my personal hobby horse) only has 4 bits of state per cell, so a 4x4 grid of cells can't count more than 2^64, no matter what. Finding the highest it could count would be interesting. For small grids, the edge connections would dominate the outcome.
There are three states: A, B, C. (States are just like goto targets in C.) State B passes control to A and C, but states A and C don't "know about" each other; they only pass control back to B. This is a sort of modular construction, whereas in true spaghetti code each state would be able to pass to all the others.
This program has some other interesting features. It never prints a blank (that is, whenever it scans a 0, it prints 1, 2, or 3). Additionally, every instruction changes either the state or the color -- there are no "lazy instructions" like B1 -> 1LB that just move position without changing anything else.