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The EU climate neutrality goal requires a strong reduction in fossil fuel use. However, whether a complete phase-out is feasible and desirable remains unclear. Using an integrated assessment model, we quantify the additional effort needed to achieve a nearly complete EU-wide phase-out of fossil fuels by 2050 compared to a least-cost net-zero scenario. We show that in the least-cost scenario fossil fuel consumption already decreases by 90% from 2020 to 2050 and is compensated by renewable power, direct electrification, as well as some hydrogen and biofuels. However, hard-to-replace oil-based hydrocarbons and natural gas persist primarily in the chemical industry, aviation and shipping. Phasing them out requires the large-scale deployment of costly carbon-neutral e-fuels, which substantially increases marginal abatement costs from 460 EUR to 630 EUR tCO2-1 (500-1000 EUR tCO2-1). Our works shows the additional transformation challenges if the EU aims to strengthen its climate policy commitment with a full fossil phase-out target.