We study the feasibility of reaching the ultrastrong (USC) and deep-strong coupling (DSC) regimes of light-matter interaction, in particular at resonance condition, with a superconducting charge qubit, also known as Cooper-Pair box (CPB). We show that by shunting the charge qubit with a high-impedance LC-circuit, one can maximally reach both USC and DSC regimes exceeding the classical upper bound ∣g∣≤ωqωr/2 between two harmonic systems with frequencies ωq and ωr. In our case, the fundamental model corresponds to an enhanced quantum Rabi model, which contains a displacement field operator that breaks its internal parity symmetry. Furthermore, we consider a multipartite device consisting of two CPBs ultrastrongly coupled to an oscillator as a mediator and study a quantum state transfer protocol between a pair of transmon qubits, all of them subjected to local incoherent noise channels with realistic parameters. This work opens the door for studying light-matter interactions beyond the quantum Rabi model at extreme coupling strengths, providing a new building block for applications within quantum computation and quantum information processing.