  The astronomical gas-phase detection of simple species and small organic
molecules in cold pre-stellar cores, with abundances as high as
$\sim$$10^{-8}-10^{-9}$ n$_\text{H}$, contradicts the generally accepted idea
that at $10$ K, such species should be fully frozen out on grain surfaces. A
physical or chemical mechanism that results in a net transfer from solid-state
species into the gas phase offers a possible explanation. Reactive desorption,
i.e., desorption following the exothermic formation of a species, is one of the
options that has been proposed. In astronomical models, the fraction of
molecules desorbed through this process is handled as a free parameter, as
experimental studies quantifying the impact of exothermicity on desorption
efficiencies are largely lacking. In this work, we present a detailed
laboratory study with the goal of deriving an upper limit for the reactive
desorption efficiency of species involved in the CO-H$_2$CO-CH$_3$OH
solid-state hydrogenation reaction chain. The limit for the overall reactive
desorption fraction is derived by precisely investigating the solid-state
elemental carbon budget, using reflection absorption infrared spectroscopy and
the calibrated solid-state band-strength values for CO, H$_2$CO and CH$_3$OH.
We find that for temperatures in the range of $10$ to $14$ K, an upper limit of
$0.24\pm 0.02$ for the overall elemental carbon loss upon CO conversion into
CH$_3$OH. This corresponds with an effective reaction desorption fraction of
$\leq$$0.07$ per hydrogenation step, or $\leq$$0.02$ per H-atom induced
reaction, assuming that H-atom addition and abstraction reactions equally
contribute to the overall reactive desorption fraction along the hydrogenation
sequence. The astronomical relevance of this finding is discussed.
