I have a circuit where I only use 3 qubits but need to utilize hundreds of classical registers. My questions are: what is the maximum number of classical bits on the current IBM Qs? What is the quantum computer with the highest number of classical bits right now that is available for researchers and the public?
The problem with your circuit is not the number of classical bits. There is no practical limit to those. The circuit that you have drawn will not run because you are doing repeated measurements on a single qubit. This is not supported on any IBM machine currently available (this would require new control electronics for operating the qubits).
Why do you need so many classical registers? One usually only uses them to catch the measurement at the end of the quantum algorithm; one for every qubit. If you do repeated measurements and intend to store statistics in separate classical registers, then you can better repeat the whole experiment. There is also on option to keep individual result of every single experiment (memory=true). Your quantum code is part of a larger whole where you mix classical and quantum computing. In the classical code (probably python) you can store any amount of data in standard facilities like lists.