Ah, thank you Eric. That helps a lot. I scrounged around and found an LM324 and threw that in the circuit. It did better, but still clamped the upper end. I see in the datasheet there's a "Input common mode voltage range" specification of Vcc-1.5V. I assume that's what you're referring to. So I connected a 5V supply to the V+ pin and now it works perfectly.
So are there op amps that will allow the full range of voltage without needing an external power supply?