Does iterated PSPACE trace equal ELEMENTARY?
Yup. If you have two algorithms $A,A'$ both in PSPACE, and you run $A'$ on the output from $A$, the resulting algorithm is in EXPSPACE.  Think about it: how much space is used by them together?  Well, it's the amount of space for $A$, plus the amount of space for $A'$.  If the input is length $n$, then the output of $A$ is at most $2^{p(n)}$, so the amount of space used by $A'$ is $q(2^{p(n)}$, where $p,q$ are polynomials.  Thus the computation is in EXPSPACE.

Also, any computation in EXPSPACE can be expressed in this way.  Suppose you have any language that can be computed in EXPSPACE, i.e., in $2^{p(n)}$ space.  Then you can take $A$ to copy its input to the output, then output a string of $2^{p(n)}$ 1's; then $A'$ can do the computation itself, and since the input to $A'$ has length $2^{p(n)}$, $A'$ will use space polynomial (in fact linear) in its input length.

You are also correct about the iteration of three of these; the result is in 2-EXPSPACE.