Well, I assume that the limitations require that no invasive procedure is permitted, although it is theoretically possible to obtain a wealth of information from a single needle: blood pressure, oxygenation and glucose levels presence of certain antigens, toxins, cellular by-products, now even a DNA profile with a new AoC. Precise EEG and EKG measurements require physical contact.
With current technology, that leaves Doppler ultrasound, thermal imaging, pulse oximetry and theoretically some sort of point-source MR imaging or penetrating radar. There are also new odor sensing chips available that are being used to diagnose bacterial infections.
I think it's GE that now has the handheld ultrasound unit, but I'm sure that advances in SoC tech may take that further by using some sort of successive approximation algorithm to artificially enhance imagery. I've been trying to wrap my head around how a point-source MRI device would work, but I kinda wondered if the techniques used in the back-scatter x-ray devices currently in use for airport security would be helpful. What would be useful is a way to simultaneously combine the output of various sensory devices to obtain useful diagnostic images.
I just keep remembering one of those weird super-vivid dreams I had in 1979. I was an army medic and I was attending to a number of victims in triage and each one of them had a wireles-link vital signs monitor velcro-strapped to their forearm about the size of a modern day smart-phone that had a single-needle probe inserted into the cephalic vein.
Weird, especially for a high-schooler, but what's weirder is how quickly technology has progressed to the point where such a device is entirely feasible.