The recent discovery of graphene(1-3) has led to many advances in two-dimensional physics and devices(4,5). The graphene devices fabricated so far have relied on SiO2 back gating(1-3). Electrochemical top gating is widely used for polymer transistors(6,7), and has also been successfully applied to carbon nanotubes(8,9). Here we demonstrate a top-gated graphene transistor that is able to reach doping levels of up to 5x10(13) cm(-2), which is much higher than those previously reported. Such high doping levels are possible because the nanometre-thick Debye layer(8,10) in the solid polymer electrolyte gate provides a much higher gate capacitance than the commonly used SiO2 back gate, which is usually about 300 nm thick(11). 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.