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 EJ/EC 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 ef<3×10−3 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 T2E for the higher levels is close to the limit of T1 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-EJ/EC transmon is a powerful tool for exploring excited states in circuit quantum electrodynamics.[/expand]