In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time an edge opens provided neither of its end vertices is part of an infinite open cluster; in the opposite case it freezes. Aldous (Math. Proc. Cambridge Philos. Soc. 128 (2000) 465–477) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay (Ann. Appl. Probab. 15 (2005) 1047–1110), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton–Watson tree that has nice scale invariant properties.