We report a field-portable lensfree microscope that can image dense and connected specimens with sub-micron resolution over a large field-of-view of 30 mm2 (i.e., 6.4 mm × 4.6 mm) using pixel super-resolution and iterative phase recovery techniques. Weighing 122 grams withdimensions of 4 cm × 4 cm × 15 cm, this microscope records lensfree in-line holograms of specimens onto an opto-electronic sensor-array usingpartially coherent illumination. To reconstruct the phase and amplitude images of dense samples (with >0.3 billion pixels in each image, i.e., >0.6 billion pixels total), we employ a multi-height imaging approach, where by using a mechanical interface the sensor-to-sample distance isdynamically changed by random discrete steps of e.g., 10 to 80 µm. By digitally propagating back and forth between these multi-height super-resolved holograms (corresponding to typically 2–5 planes), phase and amplitude images of dense samples can be recovered without the need for any spatial masks or filtering. We demonstrate the performance of this field-portable multi-height lensfree microscope by imaging Papanicolaou smears (also known as Pap tests). Our results reveal the promising potential of this multi-height lensfree computational microscopy platform for e.g., pathology needs in resource limited settings.