For the Clear Board stamp, there is no direct game rule requirement for to get all the XP before using a single health scroll.
In the current v1.1.18 version of the game, there will always be three +5XP treasure chests in the game.
The theoretical latest that one would have to use a health scroll is when they run out of HP with 11 heart containers. This playthrough would lead to 13 HP left at the end with a max clear. I've never got that much, my record is 10 HP left at the end. So the game is quite generous with extra health for the Clear Board achievement.
The gnome +9XP is very late game, getting that before having to use a health scroll would require tremendous luck. I typically get the gnome XP about the time I have used one or two health scrolls.
Typically I get the egg and one, or if lucky, two +5 XP chests before having to use a health scroll, and do so at about 8 health containers.
The mine king kill is generally easy to get immediately upon reaching lv10, to maximize the health gain there.