Carbon-coated anatase TiO$_2$ nanoparticles are a promising anode material for sodium-ion batteries, theoretically providing a satisfactory capacity combined with a relatively low cost, environmental friendliness, and high-rate capability. Nondestructive, depth-resolved hard X-ray photoelectron spectroscopy (HAXPES) is exploited to reveal the composition profile of the solid electrolyte interphase (SEI) that forms during the first sodiation (discharge) cycle and that has been determined to be (on average) 6 nm thick. Sodium chloride (NaCl), sodium fluoride (NaF), sodium carbonate (Na$_2$CO$_3$), sodium alkyl carbonates (NaCO$_3$-R), poly(ethylene oxide) (PEO), sodium hydroxide (NaOH), sodium ions, and hydrocarbons have been identified as the major species within the SEI, with a higher hydrocarbon concentration near the SEI/electrode interface and a higher sodium-ion concentration at the SEI surface. These findings give detailed insights into the complex interplay taking place at the electrolyte/TiO$_2$ nanoparticle interface during the sodiation/desodiation (discharging/charging) processes, paving the way for a deliberate optimization.