Damn, it, the Fine Structure constant! How could I let this slip past me? Basically a good chunk of reality, including the entirety of the visible and tangible world, hinge on the fact that the electromagnetic coupling constant (regulating the scale to which two charged particles interact) is 1/137. Change it to 1/1000 and we don't exist. Change it to 1 and the whole universe changes face radically.
There is no reason why the FSC (generally referred to as "alpha") has the value it has. Indeed, the value is not even that constant, but that's not a problem I should discuss here. There is however a different line of thought: any vastly different value of alpha produces a universe that doesn't have any observer within it. We, as human and observing agents, can exist only in a relatively thin range of values for alpha. So mind-boggling doubt on why nature should shape in a way that allows us to observe is upside-down: given that we exist, the universe must necessarily have the fundamental physics to allow our existence, however unlikely it would seem. This is also solved easily in the theory of multiverses: if an infinity of universes exist, each with random physical constants, only those with values that allow life will have observers inside wondering why they are so special. Nobody will exist in the others.
So yeah, 137 is a pretty important number!