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This article studies self-stabilization in networks of anonymous, asynchronously interacting nodes where the size of the network is unknown. Constant-space protocols are given for Dijkstra-style round-robin token circulation, leader election in rings, two-hop coloring in degree-bounded graphs, and establishing consistent global orientation in an undirected ring. A protocol to construct a spanning tree in regular graphs using O(log D) memory is also given, where D is the diameter of the graph. A general method for eliminating nondeterministic transitions from the self-stabilizing implementation of a large family of behaviors is used to simplify the constructions, and general conditions under which protocol composition preserves behavior are used in proving their correctness.