Here is a proof that the maximal path cannot touch every square:

I'll use this ascii diagram to refer to the squares:

|1 |2 |3 |4 |5 |6 |
|7 |8 |9 |10|11|12|
|13|14|15|16|17|18|
|19|20|21|22|23|24|
|25|26|27|28|29|30|

first, i claim that the first three squares chosen are 2,14,17. the only point of contest here is if number two is 14 or 4. assume it's 4. then later in the sequence, when square 13 is hit, 18 must follow it. thus, when square 1 is hit, at some other time, it must go to square 6, which cannot go to 18, so must go to 4, so square 4 must follow square six, not square 2. thus, the third square chosen must be square 17.

ok. now the sequence starting with square 29 must be (29,27,3), and the sequence starting with square 13 must be either (13,18,15,3) or (13,18,15,27,3), both dependent on square 17 already being hit. thus, because squares 29 and 13 must be hit sometime, then square 3 must be hit twice (or square 17 must be hit twice).

i cheat.

oberon:~$ grep ufa /usr/dict/words
manufacture