The recent discovery of graphene has led to many advances in two-dimensional physics and devices. The graphene devices fabricated so far have relied on SiO(2) back gating. Electrochemical top gating is widely used for polymer transistors, and has also been successfully applied to carbon nanotubes. Here we demonstrate a top-gated graphene transistor that is able to reach doping levels of up to 5x1013 cm-2, which is much higher than those previously reported. Such high doping levels are possible because the nanometre-thick Debye layer in the solid polymer electrolyte gate provides a much higher gate capacitance than the commonly used SiO(2) back gate, which is usually about 300 nm thick. In situ Raman measurements monitor the doping. The G peak stiffens and sharpens for both electron and hole doping, but the 2D peak shows a different response to holes and electrons. The ratio of the intensities of the G and 2D peaks shows a strong dependence on doping, making it a sensitive parameter to monitor the doping.