  We investigate the limits of thermometry using quantum probes at thermal
equilibrium within the Bayesian approach. We consider the possibility of
engineering interactions between the probes in order to enhance their
sensitivity, as well as feedback during the measurement process, i.e., adaptive
protocols. On the one hand, we obtain an ultimate bound on thermometry
precision in the Bayesian setting, valid for arbitrary interactions and
measurement schemes, which lower bounds the error with a quadratic
(Heisenberg-like) scaling with the number of probes. We develop a simple
adaptive strategy that can saturate this limit. On the other hand, we derive a
no-go theorem for non-adaptive protocols that does not allow for better than
linear (shot-noise-like) scaling even if one has unlimited control over the
probes, namely access to arbitrary many-body interactions.
