  The temperature of a bosonic/fermionic environment can be measured by
coupling a fully characterised $N$-dimensional probe to it. While
prepare-measure-reset strategies offer optimal thermometry precision, they
overlook the required time for the preparation and reset, and require excessive
control of the probe at all times. Continuously monitored probes are more
practical in this sense, as they take into account finite-time limitations.
Thus, we study the ultimate limits and the optimal structure of continuously
monitored $N$-dimensional thermometers. With the local estimation scheme our
figure of merit is the Fisher information, which inversely bounds the mean
square error. We provide an optimal strategy for both fermionic and bosonic
environments. Under reasonable assumptions it turns out that the optimal
thermometer is an effective two-level system, with a degeneracy of the ground
state that increases with $N$ -- contrary to the optimal thermometers at
equilibrium that have a single ground state degeneracy. The optimal gap also
differs from the equilibrium case, as it depends on the bath type
(fermionic/bosonic) and the specific spectral density. For $N\gg 1$, the Fisher
information can grow linearly with $N$ regardless of bath type, significantly
improving the well-known $\log^2 N$ scaling for equilibrium thermometry.
Another remarkable observation is that the scaling with $N$ does not vanish in
presence of prior ignorance, i.e., in a Bayesian setup even non-adaptive
strategies can lead to an estimation error that scales with $1/N$. In
comparison, a no-go theorem prohibits the ultimate equilibrium scaling
$1/\log^2 N$ without adaptive strategies.
