hmmm.py

import sys
sys.setrecursionlimit(20000)

stack = [[[]]]
for c in input().strip('^$'):
    if    c == '(':  stack.append([[]])    # new fork
    elif  c == '|':  stack[-1].append([])  # new tine
    elif  c == ')':  fork = stack.pop(); stack[-1][-1].append(fork)
    else:            stack[-1][-1].append(c)
[[maze]] = stack

def furthest(maze):
    if maze == []: return 0
    if isinstance(maze[0], str): return 1 + furthest(maze[1:])
    tines, *rest = maze
    if tines[-1] == []: # dead end
        return max(max(furthest(t) // 2 for t in tines), furthest(rest))
    else:
        assert rest == []
        return max(furthest(t) for t in tines)

print(furthest(maze))