Wetting of liquid droplets on passive surfaces is ubiquitous in our daily lives, and the governing physical laws are well understood. When surfaces become active, however, the governing laws of wetting remain elusive. Here, we propose chemically active wetting as a class of active systems where the surface is active due to a binding process that is maintained away from equilibrium. We derive the corresponding nonequilibrium thermodynamic theory and show that active binding fundamentally changes the wetting behavior, leading to steady, nonequilibrium states with droplet shapes reminiscent of a pancake or a mushroom. The origin of such anomalous shapes can be explained by mapping to electrostatics, where pairs of binding sinks and sources correspond to electrostatic dipoles along the triple line. This is an example of a more general analogy, where localized chemical activity gives rise to a multipole field of the chemical potential. The underlying physics is relevant for cells, where dropletforming proteins can bind to membranes accompanied by the turnover of biological fuels.