We present the deepest constraints yet on the median rest-UV+optical spectral energy distribution (SED) of z ∼ 10 galaxies prior to James Webb Space Telescope science operations. We constructed stacks based on four robust J 125 dropouts, previously identified across the GOODS fields. We used archival Hubble Space Telescope/Wide Field Camera 3 data and the full-depth Spitzer/IRAC mosaics from the GREATS program, the deepest coverage at ∼3-5 μm to date. The most remarkable feature of the SED is a blue IRAC [3.6]-[4.5] = −0.18 ± 0.25 mag color. We also find a nearly flat H 160 − [3.6] = 0.07 ± 0.22 mag color, corresponding to a UV slope β = −1.92 ± 0.25. This is consistent with previous studies and indicative of minimal dust absorption. The observed blue IRAC color and SED fitting suggest that z ∼ 10 galaxies have very young (few × 10 Myr) stellar populations, with 80% of stars being formed in the last ≲160 Myr (2σ). While an exciting result, the uncertainties on the SED are too large to allow us to place strong constraints on the presence of a nebular continuum in z ∼ 10 galaxies (as might be suggested by the blue [3.6]-[4.5] < 0 mag color). The resulting sSFR is consistent with the specific accretion rate of dark matter halos, indicative of a star formation efficiency showing quite limited evolution at such early epochs.