The discovery of the quantised Hall effect, and its subsequent topological explanation, demonstrated the important role topology can play in determining the properties of quantum systems. This realisation led to the development of topological band theory, where, in addition to band index and quasimomentum, Bloch bands are also characterised by a set of topological invariants. This topological theory can be readily extended to periodically-driven systems. In the limit of fast driving, the topology of the system can still be captured by the topological invariants used to describe static systems. In the limit of slow driving, however, situations can arise where standard topological invariants are zero, but yet, topologically protected edge modes are still observed. These "anomalous" topological edge modes have no static analogue, and are associated with a distinct topological invariant, which takes into account the full time-evolution over a driving period. Here we demonstrate the first experimental observation of such anomalous topological edge modes in an ultrafast-laser-inscribed photonic lattice. This inscription technique allows one to address each bond of a lattice independently and dynamically, generating a rich band structure with robust anomalous chiral edge modes and the potential for perfectly localised bulk states.