Even if it were practically possible (the practical challenges with how far away the nearest black holes are were outlined in other replies), such an experiment would tell us little that we don’t already know, and mostly would allow us to confirm the predictions of General Relativity for the physics outside of the event horizon, which we can already probe in a less controlled way by just watching stuff fall in from afar.
It is unfortunately impossible, given our current theories, to probe the inside of the event horizon and all the interesting physics that we do not know about that might be going on inside there, like quantum gravitational effects. This is because even if you can throw a probe in, no signal from the probe can ever come out, so you cannot gather any data.
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