Some constants are well-known.
It's a tie, in a sense. One often also has heard of the Feigenbaum constants, which preside over period-doubling bifurcations enroute to chaos. What's most interesting to me, however, are the constants which are not so well-known. They are mostly scattered in the mathematics literature, making cross-comparisons difficult. When fitting a piecewise smooth function f with a truncated Fourier series, there is always strange oscillatory behavior in the vicinity of a jump discontinuity.