Israel is a relatively rich middle-eastern democracy. Jordan should probably count too (it has its problems, yes, but so do most of the tiger states). Qatar too, to a much lesser extent and esp. if you prioritize wealth/stability over democracy per se. And Saudi Arabia is at least stable and wealthy, if not ticking off the other boxes.

But it's also worth pointing out that outside of the west and regions where the local population was so thoroughly decimated as to make the population effectively European (e.g., US and Canada), free market secular democracies are pretty rare.