  Aims. The formation and properties of star clusters at the edge of H II
regions are poorly known, partly due to limitations in angular resolution and
sensitivity, which become particularly critical when dealing with extragalactic
clusters. In this paper we study the stellar content and star-formation
processes in the young N159W region in the Large Magellanic Cloud.
  Methods. We investigate the star-forming sites in N159W at unprecedented
spatial resolution using JHKs-band images obtained with the GeMS/GSAOI
instrument on the Gemini South telescope. The typical angular resolution of the
images is of 100 mas, with a limiting magnitude in H of 22 mag (90 percent
completeness). Photometry from our images is used to identify candidate young
stellar objects (YSOs) in N159W. We also determine the H-band luminosity
function of the star cluster at the centre of the H II region and use this to
estimate its initial mass function (IMF).
  Results. We estimate an age of 2 + or - 1 Myr for the central cluster, with
its IMF described by a power-law with an index of gamma = - 1.05 + or - 0.2 ,
and with a total estimated mass of 1300 solar mass. We also identify 104
candidate YSOs, which are concentrated in clumps and subclusters of stars,
principally at the edges of the H II region. These clusters display signs of
recent and active star-formation such as ultra-compact H II regions, and
molecular outflows. This suggests that the YSOs are typically younger than the
central cluster, pointing to sequential star-formation in N159W, which has
probably been influenced by interactions with the expanding H II bubble.
