  The small semi-major axes of Hot Jupiters lead to high atmospheric
temperatures of up to several thousand Kelvin. Under these conditions,
thermally ionised metals provide a rich source of charged particles and thus
build up a sizeable electrical conductivity. Subsequent electromagnetic
effects, such as the induction of electric currents, Ohmic heating, magnetic
drag, or the weakening of zonal winds have thus far been considered mainly in
the framework of a linear, steady-state model of induction. For Hot Jupiters
with an equilibrium temperature $T_{eq} > 1500$ K, the induction of atmospheric
magnetic fields is a runaway process that can only be stopped by non-linear
feedback. For example, the back-reaction of the magnetic field onto the flow
via the Lorentz force or the occurrence of magnetic instabilities. Moreover, we
discuss the possibility of self-excited atmospheric dynamos. Our results
suggest that the induced atmospheric magnetic fields and electric currents
become independent of the electrical conductivity and the internal field, but
instead are limited by the planetary rotation rate and wind speed. As an
explicit example, we characterise the induction process for the hottest
exoplanet, KELT-9b by calculating the electrical conductivity along atmospheric
$P-T$-profiles for the day- and nightside. Despite the temperature varying
between 3000 K and 4500 K, the resulting electrical conductivity attains an
elevated value of roughly 1 S/m throughout the atmosphere. The induced magnetic
fields are predominately horizontal and might reach up to a saturation field
strength of 400 mT, exceeding the internal field by two orders of magnitude.
