An aircraft system includes a turbine engine and an air system. The turbine engine includes a flowpath, a compressor section, a combustor section and a turbine section. The flowpath extends through the compressor section, the combustor section and the turbine section from an airflow inlet into the flowpath to a combustion products exhaust from the flowpath. The combustor section includes a combustor. The air system includes an air circuit, a first boost compressor, a second boost compressor and a heat exchanger. A circuit inlet into the air circuit is fluidly coupled to the flowpath upstream of the combustor. The air circuit projects out from the circuit inlet and extends through the first boost compressor, the second boost compressor and the heat exchanger. The first boost compressor and the second boost compressor are configurable upstream of the heat exchanger along the air circuit and independently operable.