Quantum many-body systems driven far-from-equilibrium display a rich set of universal phenomena, which can generalize the notion of equilibrium phases. In this talk, we will discuss how such dynamic phases arise when quasiperiodically driving an isolated interacting many-body system with two or more incommensurate frequencies. We can make precise a notion of multiple emergent time-translation symmetries, which allows the construction of phases that fundamentally cannot be realized in static or periodically driven systems. This includes time-quasicrystals and time-translation symmetry protected topological phases. For these new phases to meaningfully exist, they must resist unbounded heating for at least a parametrically long time. We establish regimes for which this is the case and, along the way, derive an analytical description of the preheating regime for high-frequency quasiperiodic driving. Finally, we discuss a recent experimental implementation of a time-translation symmetry protected topological phase on a trapped-ion quantum processor. Relying solely on the emergent time-translation symmetries provides a more stable protection of the topological edge states against coherent errors as compared to edge state protection in phases build on an approximate microscopic symmetry.