Self-similarity and fractals have fascinated researchers across various disciplines. In graphene placed on boron nitride and subjected to a magnetic field, self-similarity appears in the form of numerous replicas of the original Dirac spectrum, and their quantization gives rise to a fractal pattern of Landau levels, referred to as the Hofstadter butterfly. Here we employ capacitance spectroscopy to probe directly the density of states (DoS) and energy gaps in this spectrum. Without a magnetic field, replica spectra are seen as pronounced DoS minima surrounded by van Hove singularities. The Hofstadter butterfly shows up as recurring Landau fan diagrams in high fields. Electron-electron interactions add another twist to the self-similar behaviour. We observe suppression of quantum Hall ferromagnetism, a reverse Stoner transition at commensurable fluxes and additional ferromagnetism within replica spectra. The strength and variety of the interaction effects indicate a large playground to study many-body physics in fractal Dirac systems.