What is the context of your question ? In introductory physics there are hardly any dimensionless physical constants, apart from ratios.
Are you sure you don't mean coefficients (e.g. for physical properties, friction coefficients and such) ?
If we leave out a whole lot of ratios (mostly dimensionless for obvious reasons),
when I look at a list , all I see are physical constants with a dimension -- apart from a few very fancy ones such as ##\alpha## and g-2 .