One of the oddest board games ever made must be Pinhead, released by Remco in 1959.
The first inkling of strangeness comes right away, as you peruse the cover. Why is a "game of hide and seek" called "Pinhead"? What does a deforming neurodevelopmental disorder have to do with a classic children's playtime activity? Why is the one boy so much bigger than the other children? Is it also about dwarfism and/or gigantism?
You might expect some answers to these baffling questions once you open the box, but no: the oddness just intensifies. There is, in fact, a pinhead on the house shaped game board, and he is "hiding" out in the open, in the middle of what appears to be a hallway. So...not hiding, then. I mean, wouldn't it have been more like hiding if he was tucked away in the attic clutter, or stuck behind one of the basement appliances??
Remco games were notable for: 1. being strange, and 2. having unusual methods of rolling the dice. Remco's "Tumblebum Dice Games" included an hourglass shaped device with dice inside. Tipping the device over essentially rolled the dice. Pinhead features a different mechanism: a dice box, in which the dice are shaken while the lid is closed. These elaborate dice rolling devices seem to have been Remco's attempt to enliven games that were otherwise rather simple, straightforward "tracks", wherein players simply moved their markers along a course. Remco games are relatively scarce, and strange though they may be, are worth snapping up when found.