  Axions play a central role in inflationary model building and other
cosmological applications. This is mainly due to their flat potential, which is
protected by a global shift symmetry. However, quantum gravity is known to
break global symmetries, the crucial effect in the present context being
gravitational instantons or Giddings-Strominger wormholes. We attempt to
quantify, as model-independently as possible, how large a scalar potential is
induced by this general quantum gravity effect. We pay particular attention to
the crucial issue which solutions can or cannot be trusted in the presence of a
moduli-stabilisation and a Kaluza-Klein scale. An important conclusion is that,
due to specific numerical prefactors, the effect is surprisingly small even in
UV-completions with the highest possible scale offered by string theory.
  As we go along, we discuss in detail Euclidean wormholes, cored and extremal
instantons, and how the latter arise from 5d Reissner-Nordstrom black holes. We
attempt to dispel possible doubts that wormholes contribute to the scalar
potential by an explicit calculation. We analyse the role of stabilised
dilaton-like moduli. Finally, we argue that Euclidean wormholes may be the
objects satisfying the Weak Gravity Conjecture extended to instantons.
