User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-31580368-20190630051001

"Moreover, there will be no real examples between level 52 and 60, because all of them are completely ill-defined".

53. finite promise games (upper bound)

56. Friedman's finite trees (upper bound)

58. TON with Fundamental Sequences (upper bound)

59. Greedy clique sequence (upper bound)

60. Laver table (upper bound)

PS: I believe that the question of what a "ill-defined function" means is to be attributed to the question of the philosophy of mathematics. For example: any of our functions using ω is "ill-defined function" in terms of ultrafinitist.