We investigate the influence of dynamical charm quarks on observables that depend explicitly on the charm quark fields, like the pseudo-scalar and vector masses $m_{\eta_c}$ and $m_{J/\psi}$, the hyperfine splitting $(m_{J/\psi} − m_{\eta_c} )/m_{\eta_c}$, the charm quark mass and the meson decay constants. For this purpose, instead of working in full QCD we study a simplified setup. We simulate two theories: $N_f = 0$ QCD and QCD with $N_f = 2$ dynamical quarks at the charm mass. The absence of light quarks allows us to reach extremely fine lattice spacings (0.02 fm < a < 0.05 fm) which are crucial for reliable continuum extrapolations. Our main result is a comparison of various quantities in the continuum limit. For the hyperfine splitting we find that the effects of a dynamical charm quark are below our statistical precision of 2%.