High charge carrier mobilities play a fundamental role for high-frequency electronics, integrated optoelectronics as well as for sensor and spintronic applications, where device performance is directly linked to the magnitude of the carrier mobility. Van der Waals heterostructures formed by graphene and hexagonal boron nitride (hBN) already outperform all known materials in terms of room temperature mobility. Here, we show that the room temperature carrier mobility of today's best graphene/hBN devices can be surpassed by more than a factor of three by heterostructures formed by tungsten diselenide (WSe₂), graphene and hBN, which can have mobilities as high as 350,000 cm²/(Vs) and resistivities as low as 15 Ohm. The resistivity of these devices shows a significantly weaker temperature dependence than the one of graphene on any other known substrate. Notably, the reduced temperature dependence and the resulting mobility enhancement in graphene show an unexpected relation to the thickness of the WSe₂ layer and to the minimum conductivity at the charge neutrality point, suggesting a role of increased charge disorder and/or enhanced screening and questioning the current understanding of electron-phonon scattering in graphene-based van der Waals heterostructures.