All of the things I keep in my place are just for measuring gains and constraints. Tape measures, calipers (not bioimpedance), and a scale can tell you quite a bit, especially if you know how to use them and their accuracies. Measuring your heart rate is important. Rapid increases or decreases in waking heart rate are thought to be symptomatic of overtraining. But a monitor? Mmm. Just use a watch unless you've got nothing else more pressing to spend on. Mirrors are good and all - motivation and feedback there. But you said they're off limits. If I had to choose only one item, it would be this:
A digital camera.
If you take the pictures in the same lighting and color backdrop, pictures over time will tell you if you are gaining or losing bodyfat and whether you are gaining or losing muscle. Unlike a mirror, you can take full frontal pictures of your back by using the delay feature on most digitals. I don't know about the rest of you, but constantly looking in the mirror just drives me bananas after a while. Take the same pictures every two weeks (use the same lighting and poses!) and you can really notice what's going on with your body over time.
Crazy idea: Fill a deep jar or pipe 75-80% full. Graduate it. Mark the water line with permanent marker. Submerge your arm in it up to some identifiable level, like say to the acromion process. Mark that. Do it again in a month, or say, at the end of an HST cycle. Calculate volume change. Volume change = cubic inch change in your arm over one month. Its measuring the displacement of your arm like they do for engines.
"Dude! You've got huge guns!"
"Thanks. They're 300 cc's."
or
"That last HST cycle really paid off. I added 18 cc's to my right arm alone."