We consider the power of objects in the unbounded concurrency shared memory model, where there is an infinite set of processes and the number of processes active concurrently may increase without bound. By studying this model we obtain new results and observations that are relevant and meaningful to the standard bounded concurrency model. First we resolve an open problem from 2006 and provide, contrary to what was conjectured, an unbounded concurrency wait-free implementation of a swap object from 2consensus objects. This construction resolves another puzzle that has eluded us for a long time, that of considerably simplifying a 16 year old complicated bounded concurrency swap construction. A further insight to the traditional bounded concurrency model that we obtain by studying the unbounded concurrency model, is a refinement of the top level of the waitfree hierarchy, the class of infinite-consensus number objects. First we resolve an open question of Merritt and Taubenfeld from 2...