  We propose and analyze two distinct routes toward realizing interacting
symmetry-protected topological (SPT) phases via periodic driving. First, we
demonstrate that a driven transverse-field Ising model can be used to engineer
complex interactions which enable the emulation of an equilibrium SPT phase.
This phase remains stable only within a parametric time scale controlled by the
driving frequency, beyond which its topological features break down. To
overcome this issue, we consider an alternate route based upon realizing an
intrinsically Floquet SPT phase that does not have any equilibrium analog. In
both cases, we show that disorder, leading to many-body localization, prevents
runaway heating and enables the observation of coherent quantum dynamics at
high energy densities. Furthermore, we clarify the distinction between the
equilibrium and Floquet SPT phases by identifying a unique micromotion-based
entanglement spectrum signature of the latter. Finally, we propose a unifying
implementation in a one-dimensional chain of Rydberg-dressed atoms and show
that protected edge modes are observable on realistic experimental time scales.
