  Thermal equilibrium states are exponentially hard to distinguish at very low
temperatures, making equilibrium quantum thermometry in this regime a
formidable task. We present a thermometric scheme that circumvents this
limitation, by using a two-level probe that does not thermalize with the sample
whose temperature is measured. This is made possible thanks to a suitable
interaction that couples the probe to the sample and to an auxiliary thermal
bath known to be at a higher temperature. Provided a reasonable upper bound on
the temperature of the sample, the resulting 'thermometric machine' drives the
probe towards a steady state whose signal-to-noise ratio can achieve values as
high as $\mathcal{O}(1/T)$. We also characterize the transient state of the
probe and numerically illustrate an extreme reduction in the number of
measurements to attain a given precision, as compared to optimal measurements
on a thermalized probe.
