Inverse design has emerged as a powerful engineering approach in nanophotonics, leading to devices with unintuitive designs that extend beyond human intuition. However, inverse-designed devices have been limited by costly and time-consuming fabrication iterations, simulation-reality gaps, and the fixed nature of etched devices. Separately, programmable photonics has emerged as an approach to modify the function of photonic devices after fabrication; recent free-form programmable-photonic devices have been scaled to have many degrees of freedom, in principle providing platforms well-suited for programmably realizing inverse design of conventional linear-photonic functions. Here we demonstrate in situ training of a single free-form programmable-photonic device to achieve a variety of functions that each would have been implemented in separate devices in the conventional inverse-design paradigm. Our platform is a thin-film lithium niobate slab waveguide with full programmability over the refractive index of the two-dimensional slab, with a tuning range of up to 1.7e-3 for each of the 10^4 effective pixels. We demonstrate 4 x 4 Gaussian-mode permutations, 4 x 4 Hermite-Gaussian mode demultiplexing, and 8 x 8 Gaussian-mode permutations, with maximum crosstalk of -27dB, -13dB, and -13dB, as well as 4 x 4 matrix-vector multiplication. We further demonstrate engineering of spectral responses, showcasing both wavelength-independent operation throughout the 1500-1600 nm window and wavelength demultiplexing between 1310 nm and 1550 nm, with maximum crosstalk of -8.4dB. Since the device is reconfigured for each function after fabrication, the investment in device fabrication, characterization, and calibration can be amortized across the various photonic functions the device is ultimately used for.