To any algebraic dynamical system, by which I mean a dominant rational selfmap of a projective algebraic variety, one can associate a sequence of dynamical degrees; these are birational conjugacy invariants of the dynamics and play a key role in the study of both complex and arithmetic dynamics. In examples where the dynamical degrees can be readily computed, they tend to be algebraic integers. However, I will explain joint work with J. Bell and D. Diller in which we construct a rational selfmap of the projective plane whose (first) dynamical degree is a transcendental number.