We demonstrate enhanced hole injection and lowered driving voltage in vacuum-deposited organic light-emitting diodes (OLEDs) with a hole-transport layer using the starburst amine 4,4',4"-tris(N,N-diphenyl-amino)triphenylamine (TDATA) p-doped with a very strong acceptor, tetrafluoro-tetracyano-quinodimethane (F-4-TCNQ) by controlled coevaporation. The doping leads to high conductivity of doped TDATA layers and a high density of equilibrium charge carriers, which facilitates hole injection and transport. Moreover, multilayer OLEDs consisting of double hole-transport layers of thick p-doped TDATA and a thin triphenyl-diamine (TPD) interlayer exhibit very low operating voltages.