The Kasner spacetime is a cosmological model of an anisotropic expanding universe without matter and an exact solution to the Einstein vacuum equations. Due to its curvature singularity (at the Big Bang), it is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this talk we will review some obstructions to continuous extendibility and prove that the Kasner spacetime is even inextendible as a Lorentzian manifold with a continuous metric. We do so by adapting the proof of the C^0-inextendibility of the maximal analytically extended Schwarzschild spacetime established by Jan Sbierski.