In this paper we present a new notion of what it means for a problem in $P$ to be inherently sequential. Informally, a problem $L$ is {\em strictly sequential P-complete\/} if when the best known {\em sequential\/} algorithm for $L$ has polynomial speedup by parallelization, this implies that {\em all\/} problems in $P$ have a polynomial speedup in the parallel setting. The motivation for defining this class of problems is to try and capture the problems in $P$ that are truly inherently sequential. Our work extends the results of Condon who exhibited problems such that if a polynomial speedup of their best known {\em parallel\/} algorithms could be achieved, then all problems in $P$ would have polynomial speedup. We demonstrate one such natural problem, namely the {\em Multiplex-select Circuit Problem\/} (MCP). MCP has one of the highest degrees of sequentiality of any problem yet defined. On the way to proving MCP is strictly sequential $P$-complete, we define an interesting model, the {\em register stack machine}, that appears to be of independent interest for exploring pure sequentiality.