Crucially, there are 16 corridors.
So with 4 players, it seems like you could map out 26×4 + 16 = 120 squares. Although I have not constructed an example of this, I think that there are enough rooms with 3 or 4 exits that this is extremely likely to be possible. How can we reveal the last four squares?
Serellia's ability to avoid traps can put face-down tiles on the remaining four squares. I don't think that counts.
Azoth's Stasis spell trivially gives you enough turns to reveal the last four. This seems like cheating.
Other unusual ways to reveal tiles are exiting from the Catacombs (but still only one tile is revealed on that turns) and finding a secret door (ditto). If all four characters are in the Treasure Chamber when the dragon wakes up, they can reveal as many as four tiles in one turn, but they each used a turn without a tile reveal getting in, so that doesn't help.
Ah now, what if Helena runs through a face-down tile that Serellia placed? Can she do that? Taking the rules literally, it is "placed face down" and Helena can run through "placed tiles", so yes! (I've added this to the rulings page.) And they are revealed when a character enters them, which Helena is doing in this case. So this could work, at least if you buy my reading of the rules. But can you actually construct a game that uses this to get a face-up tile in every square? Besides the availability of tiles with enough exits, there's the matter of getting Helena over to where Serellia is placing face-down tiles without wasting too many turns stepping on already-placed tiles. (I think it goes without saying that we are not requiring the characters to survive the game. That would make it a much more difficult challenge.)
Tile availability: Is there a set of tiles with the right exits for the characters to take advantage of this? If searching is allowed, it turns out this is trivial. Like many of the decks in the game, one of the Search cards tells you to shuffle the deck and draw another card. This means you can draw a Secret Door whenever you need one. Like the Azoth solution above, it seems boring to allow this, so let's assume no searching. The available tiles are:
You may have noticed that all of the tiles come in symmetric sets of left/right except for the 8x right, 6x left in the last category. I wondered if I were missing tiles, since my set came used on eBay. But I know my Catacombs set is complete because I punched it myself. And these pictures of the unpunched base set match my set perfectly. It would seem that Dungeonquest, like the Universe we inhabit, has a slight asymmetry built into it.
To be continued…