Summary: We investigate a nonperturbative formulation of quantum gravity defined via Euclidean dynamical triangulations (EDT) with a non-trivial measure term in the path integral. We are motivated to revisit this older formulation of dynamical triangulations by hints from renormalization group approaches that gravity may be asymptotically safe and by the emergence of a semiclassical phase in causal dynamical triangulations (CDT).We study the phase diagram of this model and identify the two phases that are well known from previous work: the branched polymer phase and the collapsed phase. We verify that the order of the phase transition dividing the branched polymer phase from the collapsed phase is almost certainly first-order. The nontrivial measure term enlarges the phase diagram, allowing us to explore a region of the phase diagram that has been dubbed the crinkled region. Although the collapsed and branched polymer phases have been studied extensively in the literature, the crinkled region has not received the same scrutiny. We find that the crinkled region is likely a part of the collapsed phase with particularly large finite-size effects. Intriguingly, the behavior of the spectral dimension in the crinkled region at small volumes is similar to that of CDT, as first reported in [the authors, “Evidence for asymptotic safety from lattice quantum gravity”, Phys. Rev. Lett. 107, No. 16, Article ID 161301, 4 p. (2011; doi:10.1103/PhysRevLett.107.161301)], but for sufficiently large volumes the crinkled region does not appear to have 4-dimensional semiclassical features. Thus, we find that the crinkled region of the EDT formulation does not share the good features of the extended phase of CDT, as we first suggested in [loc. cit.]. This agrees with the recent results of J. Ambjørn et al. [J. High Energy Phys. 2013, No. 10, Paper No. 100, 23 p. (2013; Zbl 1342.83052)], in which the authors used a somewhat different discretization of EDT from the one presented here.