When a continuous dynamical system admits a Markov partition, the exponential of its topological entropy, called the growth rate, is a weak Perron number. The Thurston set for a family of such maps is the closure of the set of Galois conjugates of growth rates of maps in the family. I will discuss some of the basic geometric and topological properties of the Thurston set and a closely related set, the Thurston Master Teapot, for the family of superattracting real quadratic polynomials. In particular, I will show that Thurston's Master Teapot is connected.