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According to the Blender docs https://docs.blender.org/manual/en/latest/render/shader_nodes/input/light_path.html ray depth gives the Number of times the ray has been reflected or transmitted on interaction with a surface.

Where do you start to count? So is the number of bounces the bounces going outward from the camera or going outward from the light source?

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Rays start from the camera. For the first object a camera ray hits, the ray depth is evaluated as 0:

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Here, a camera ray strikes the actively selected cube, which evaluates the ray depth as 0, returning full emission. Another camera ray strikes the mirror behind it, incrementing the ray depth, then reflects onto cube, which now evaluates the ray depth as 1.

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