Poisson-Delaunay triangulation is a classical random triangulation model. In it, it was shown by Bonnet and Chenavier that the maximal degree in a growing window eventually concentrates at two (slowly growing) consecutive values. We will discuss some of the ideas behind this result as well as their possible generalizations to new triangulation models — Beta- and Beta-Prime-Delaunay — introduced recently by Gusakova et al. in a series of papers. The talk is based on an ongoing project with Gilles Bonnet.