Topological phases without crystalline counterparts We construct a higher-order topological phase protected by a point group symmetry that is impossible in any crystalline system. The tight-binding model describes a superconductor on a quasicrystalline Ammann–Beenker tiling which hosts localized Majorana zero modes at the corners of an octagonal sample. The Majorana modes are protected by particle-hole symmetry and by the combination of an 8-fold rotation and in-plane reflection symmetry. We find a bulk topological invariant associated with the presence of these zero modes, and show that they are robust against large symmetry preserving deformations, as long as the bulk remains gapped. The nontrivial bulk topology of this phase falls outside all currently known classification schemes. Contents Code to generate all figures in the manuscript: in octo_hoti.ipynb and octo_hoti_kpm.ipynb. Data to generate large patches of the tiling. Data produced by the longer calculations. Code and results of symmetry analysis using Qsymm in symmetry_analysis.ipynb. Requirements kwant >= 1.4 qsymm >= 1.1