High bandwidth, uncooled, Indium Arsenide (InAs) electron avalanche photodiodes (e-APDs) with unique and highly desirable characteristics are reported. The e-APDs exhibit a 3dB bandwidth of 3.5 GHz which, unlike that of conventional APDs, is shown not to reduce with increasing avalanche gain. Hence these InAs e-APDs demonstrate a characteristic of theoretically ideal electron only APDs, the absence of a gain-bandwidth product limit. This is important because gain-bandwidth products restrict the maximum exploitable gain in all conventional high bandwidth APDs. Non-limiting gain-bandwidth products up to 580 GHz have been measured on these first high bandwidth e-APDs.