For long, we have studied tiny energy harvesters to liberate sensors from batteries. With remarkable progress in embedded deep learning, we are now re-imagining these sensors as intelligent compute nodes. Naturally, we are approaching a crossroad where sensor intelligence is meeting energy autonomy enabling maintenance-free swarm intelligence and unleashing a plethora of applications ranging from precision agriculture to ubiquitous asset tracking to infrastructure monitoring. One of the critical challenges, however, is to adapt intelligence fidelity in response to available energy to maximise the overall system availability. To this end, we present the design and implementation of ePerceptive: a novel framework for best-effort embedded intelligence, i.e., inference fidelity varies in proportion to the instantaneous energy supplied. ePerceptive operates on two core principles. First, it enables training a single deep neural network (DNN) to operate on multiple input resolutions without compromising accuracy or incurring memory overhead. Second, it modifies a DNN architecture by injecting multiple exits to guarantee valid, albeit lower-fidelity inferences in the event of energy interruption. The combination of these techniques offers a smooth adaptation between inference latency and recognition accuracy while matching the computational load to the available power budget. We report the manifestation of ePerceptive in designing batteryless cameras and microphones built with TI MSP430 MCU and off-the-shelf RF and solar energy harvesters. Our evaluation of these batteryless sensors with multiple vision and acoustic workloads suggest that the dynamic adaptation of ePerceptive can increase the inference throughput by up to 80% compared to a static baseline while ensuring a maximum accuracy drop of less than 6%.