Some good responses.
It depends a lot on the species of fish and their response to temperature and dissolved oxygen.
Dissolved oxygen concentrations are (generally) higher in cold water than warm water. Species like trout have a high demand for dissolved oxygen, and are typically found in low water temperatures (55-70 degrees F they're considered "coldwater" species). "Cool water" species like northern pike, yellow perch, and walleye like water temperatures can live in temperatures a little higher, but prefer maybe 65-75 degrees. "Warmwater" species like your typical largemouth bass, channel catfish, and bluegill, can thrive in temperatures upwards of 75 degrees. Some species are specially adapted- gar and bowfin can gulp air from the surface, topminnows and killifish cruise just under the water surface, where oxygen concentration is high. So short answer: depending on what species you're interested in, minimum depth can be a couple inches or dozens of feet.
Small ponds warm up faster (and get warmer) than small ponds, shallow ponds tend to have more oxygen than small ones (greater interaction with the atmosphere). Shallow ponds also tend to have more aquatic vegetation, which may or may not be a good thing depending on your management goals.
If you're in an ice-free (or largely ice-free) area, 4-5 feet is probably adequate for most fish species. If you get ice cover for more than a couple weeks a year, you may consider depths closer to 8-10 feet. If you're willing to run a pump during winter to keep open, ice-free areas, you could probably get away with a shallower pool.
I'm a big proponent of small, fishless pools. Lots of invertebrate and amphibian species have been declining over the past several decades due to landscape alteration and chemical spraying; thanks for incorporating some into your plan!