How would you implement a custom checked exception that requires additional context to be attached as the exception propagates up multiple layers of a call stack, without losing information at each layer?

A common pattern for preserving context across multiple call stack layers involves each layer catching the exception, adding its own relevant contextual information (like which specific record ID was being processed when the failure occurred), and rethrowing either the same exception (after mutating/adding to accumulated context, if the exception type supports that) or a new wrapping exception with the original preserved as the cause -- accumulating context this way at each layer (rather than only having the innermost, most technical error message) produces a much more actionable end-to-end error trail when the exception is eventually logged or displayed, letting an on-call engineer understand not just what technically failed but the full business-level context of what the system was attempting to do when it failed.

public class BatchProcessingException extends Exception {
    private final List<String> processingContext = new ArrayList<>();

    public BatchProcessingException(String message, Throwable cause) { super(message, cause); }

    public BatchProcessingException addContext(String context) {
        processingContext.add(context);
        return this;
    }
}

// Layer 1 (innermost): throws with initial context
// Layer 2: catches, adds "processing batch #47" context, rethrows the SAME exception instance
// Layer 3: catches, adds "nightly reconciliation job" context, logs full accumulated context chain