  Qudits provide a resource-efficient alternative to qubits for quantum
information processing. The multilevel nature of the transmon, with its
individually resolvable transition frequencies, makes it an attractive platform
for superconducting circuit-based qudits. In this work, we systematically
analyze the trade-offs associated with encoding high-dimensional quantum
information in fixed-frequency transmons. Designing high $E_J/E_C$ ratios of up
to 325, we observe up to 12 levels ($d=12$) on a single transmon. Despite the
decreased anharmonicity, we demonstrate process infidelities $e_f < 3 \times
10^{-3}$ for qubit-like operations in each adjacent-level qubit subspace in the
lowest 10 levels. Furthermore, we achieve a 10-state readout assignment
fidelity of 93.8% with the assistance of deep neural network classification of
a multi-tone dispersive measurement. We find that the Hahn echo time $T_{2E}$
for the higher levels is close to the limit of $T_1$ decay, primarily limited
by bosonic enhancement. We verify the recently introduced Josephson harmonics
model, finding that it yields better predictions for the transition frequencies
and charge dispersion. Finally, we show strong $ZZ$-like coupling between the
higher energy levels in a two-transmon system. Our high-fidelity control and
readout methods, in combination with our comprehensive characterization of the
transmon model, suggest that the high-$E_J/E_C$ transmon is a powerful tool for
exploring excited states in circuit quantum electrodynamics.
