The paper investigates the quasigroup Q_s constructed on the well-ordered set of natural numbers by placing a number $s$ known as the seed in the top left-hand corner of the body of the multiplication table, and then completing the Latin square using the greedy algorithm that chooses the least possible entry at each stage. The initial motivation comes from the theory of combinatorial games, where Q₀ gives the usual nim sum, while Q₁ gives the corresponding sums for positions in misere nim. The multiplication groups of these quasigroups are analyzed. The alternating group of the natural numbers is a subgroup of the multiplication groups. It is shown that these so-called greedy quasigroups Q_s are mutually non-isomorphic. The quasigroup Q₁ is subdirectly irreducible. For s>1, the greedy quasigroups Q_s are simple, and for s>2 they are rigid, possessing no non-trivial automorphisms. Indeed in this case the endomorphism monoid contains just the identity and a single constant. The subquasigroup structures of the Q_s are also determined. While Q₀, Q₁ have uncountably many subquasigroups, and Q₂ has just one proper, non-trivial subquasigroup, Q_s has none for s>2.