American Airlines, delayed 3+ hours: no fixed cash compensation applies.
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Not automatically for delays alone. Even at 3+ hours, US rules don't mandate cash compensation for delays — though if the cause was within the airline's control (a maintenance issue, not weather), most major carriers commit to free rebooking, meals, and hotel or ground transport under DOT customer service plans.
If your flight is cancelled, you're legally entitled to a full, automatic cash refund for the unused portion of your ticket if you choose not to travel, plus free rebooking — but there's no extra compensation payment like in the EU.
You may be owed up to 400% of your one-way fare (capped, doubled if you're rebooked very late). This is the one case where US rules do require cash compensation — if you're involuntarily bumped from an oversold flight, not for ordinary delays or cancellations.
American Airlines is a US carrier, so US DOT rules apply rather than the EU's EU261 regulation.