We consider the space of complete minimal surfaces in $H^3$ with a (free) boundary at infinity. We explain how the Willmore energy is a natural functional on this space. We study the possible loss of compactness in the space of such surfaces with energy bounded above. This question has been extensively studied for various energies in the context of closed surfaces, starting with the classical work of Sacks and Uhlenbeck on harmonic maps. We derive analogues of epsilon-reglarity, removability of singularities and bubbing in this setting. A key difference (and difficulty) compared to the classical picture is a lack of energy quantization. Joint with R. Mazzeo.