Related to my answer on this question, I'm not sure of a detail.
Assume you have a Turing Machine which simulates all possible Turing Machines all at once (meaning it does not "page" its data, i.e. it can't erase data from one "process" to write data from another "process"; it must keep all "processes" "in memory" at the same time). Then, this Turing Machine must simulate itself, because it is a Turing Machine and the definition of this machine is that it simulates all possible Turing Machines. It must also simulate the simulation of itself, and the simulation of that, and so on, infinitely recursive. Therefore, this Turing Machine must have infinite data on its tape.
My question is, can this Turing Machine exist, and why or why not?