ICOT – 27 is definitely not a possibility. Remember that on day 27 he starts out 26 feet high and makes it only to 29 before falling back to 27. Perhaps you meant to say it could be 28 or 29. That would be true, since it is ambiguous what will happen at the lip. The answer I gave (28) assumes that reaching the lip prevents the snail from falling back down. Good point though!
jfem: I am a liar is a statement that cannot be made by either a patent liar or a patent truth teller. Only someone who sometimes lies and sometimes tells the truth can say that. Not so the statement “My father’s name is Daniel”, which can be stated by either a truth teller or a liar (and its validity can be discovered).
Even so, IMO “I am a liar” does not work because a liar could say that (even though it is the truth – someone who sometimes lies is still a liar).