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(La and Ga)-doped tin monoxide (stannous oxide, tin (II) oxide, SnO) thin films were grown by plasma-assisted and suboxide molecular beam epitaxy with dopant concentrations ranging from $\approx5\times10^{18}$cm$^{-3}$ to $2\times10^{21}$cm$^{-3}$. In this concentration range, the incorporation of Ga into SnO was limited by the formation of secondary phases observed at $1.2\times10^{21}$cm$^{-3}$ Ga, while the incorporation of La showed a lower solubility limit. Transport measurements on the doped samples reveal that Ga acts as an acceptor and La as a compensating donor. While Ga doping led to an increase of the hole concentration from $1\times10^{18}$cm$^{-3}-1\times10^{19}$cm$^{-3}$ for unintentionally (UID) SnO up to $5\times10^{19}$cm$^{-3}$, La-concentrations well in excess of the UID acceptor concentration resulted in semi-insulating films without detectable $n$-type conductivity. Ab-initio calculations qualitatively agree with our dopant assignment of Ga and La, and further predict In$_\text{Sn}$ to act as an acceptor as well as Al$_\text{Sn}$ and B$_\text{Sn}$ as donor. These results show the possibilities of controlling the hole concentration in $p$-type SnO, which can be useful for a range of optoelectronic and gas-sensing applications.