Topologically　protected　edge　states　exactly　at　topological　phase　boundaries　challenge　the　conventional　belief　that　topological　states　must　be　associated　with　a　bulk　energy　gap.　Because　periodically　driven　(Floquet)　systems　host　unusually　intricate　topological　phase　boundaries,　topological　edge　states　can　be　prolific　at　such　Floquet　quantum　criticality.　Working　on　a　class　of　chiral-symmetric,　Floquet-driven　Majorana　fermion　chains,　we　analytically　and　computationally　show　that　the　precise　boundaries　between　different　Floquet　topological　gapped　phases　can　accommodate　topological　edge　modes,　including　the　so-called　critical　Majorana　pi　modes.　We　also　identify　a　general　bulk-edge　correspondence　formula　to　predict　and　understand　the　emergence　of　topological　edge　modes　at　Floquet　quantum　criticality.　Of　direct　interest　to　quantum　simulation　experiments,　our　work　opens　an　avenue　for　exploring　topological　physics　intrinsic　to　nonequilibrium　phase　boundaries.