In modern operating systems and programming languages adapted to multicore computer architectures, parallelism is abstracted by the notion of execution threads. Multi-threaded systems have two major specificities: on the one part, new threads can be created dynamically at runtime, so there is no bound on the number of threads participating in long-running executions. On the other part, threads have access to a memory allocation mechanism that cannot allocate infinite arrays. These specificities make it challenging to adapt some algorithms to multi-threaded systems, in particular those that need to assign one shared register per process. This paper explores the synchronization power of shared objects in multi-threaded systems by extending the famous Herlihy's wait-free hierarchy to take these constraints into consideration. It proposes to subdivide the set of objects with an infinite consensus number into nine new degrees, depending on their ability to synchronize a bounded, finite or infinite number of processes, with or without the need to allocate an infinite array. To show the relevance of the proposed extension, for each new degree it is either proved that it is empty, or an object illustrating it is proposed.