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Driving a laser-plasma accelerator (LPA) with an incoherent superposition (stack) of multi-TW optical blocks of different colors opens the way to GeV-scale electron acceleration in a mm-length, dense plasma (n₀ ∼ 10¹⁹ cm^-3), without resorting to single-shot PW facilities. Stack-driven LPA affords production of electron beams (e-beams) at a kHz repetition rate, at a manageable average laser power. Immunity of the stack to self-compression (i) avoids deformation (elongation) of the accelerating bucket (a ``bubble'' of electron density), keeping the e-beam background-free through dephasing, and (ii) delays dephasing, thereby doubling electron energy against the predictions of standard scaling.  In our PIC simulations, quasi-monoenergetic e-beams (or trains of such beams, in a single shot) have a 5-D brightness 10¹⁶ - 10¹⁷ A/m² and mean energy tunable up to 1 GeV without changing the target. These unconventional beams, inaccessible with standard acceleration techniques, emit highly collimated, quasi-monochromatic, gigawatt γ-ray pulses via Thomson scattering process, in the range 3 - 17 MeV, each pulse corresponding to a distinct energy band (with ∼ 10⁶ photons per band) [S. Y. Kalmykov et al., New J. Phys. 20, 023047 (2018).]