Not really a question, as I solved it... but it was a real hum-dinger!
I had an issue very similar to this thread wherein a single MAC address (HP) was exhausting my DHCP scope at remote locations, just cycling through every single possible IP address--a veritable Goldilocks if you will. During high-traffic times this would cause packet loss on the entire small router due to the CPU overhead of servicing so many requests.
After months and months we found the culprit: iPads... connected to an HP wireless bridge (thus the HP MAC address). After a certain Apple iOS update, iOS devices would determine if wifi was working by trying to reach an apple.com URL and, if unsuccessful, would start a reconnect and request a new DHCP IP address.
Well, we had a whitelist filter, and apple.com was most definitely not on the whitelist.
This iOS behaviour was not configurable, we ended up having to whitelist apple.com.
So summary: DHCP scope being exhausted by an HP MAC address.
Culprit: iOS devices connected to HP wireless bridge were requesting new DHCP addresses when they could not reach apple.com.
Solution: Allow access to apple.com
Alternate Solution: Follow Cisco's advice and "Use non-Apple devices." :P