What would be an effective BJT transistor configuration for implementing an amplifier with a current gain of greater than 1000? This can be a multistage configuration, but it may only use BJT transistors. I was thinking of using a common-collector configuration for such a purpose. What would be appropriate biases for the resistors in the amplifiers (along with capacitor values if there any worth noting)?
As mentioned in the comments to your question, there are two simple beta-boosting configurations: Darlington and Sziklai.
For both configurations the effective beta (current gain) is a multiplication of the betas of the two transistors. Below are the schematics for these two configurations.
pay attention that the resistor in both cases is not necessary but it helps to improve the slow response time of these configurations and prevents leakage current through Q1 from biasing Q2 into conduction. Also make sure to choose its value so as not to steal substantial base current from Q2.