I just opened my oil filler cap while the engine was running, just to see what's what, and I noticed that the timing chain is oscillating about ±3mm back and forth on the sprocket.

Is this a normal/permissible amount of chain wander, or is this a heaven-sent warning of imminent timing chain demise?

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RIP:
2011 E350 BlueTEC, Schwartz 81k (totaled by uninsured drunk driver)
1964 Unimog 404.1 (wrongfully towed by the city)
1994 E320, Brilliantsilber 208k (transmission shuffled off after sudden catastrophic fluid loss)
1982 300D, Silberblau under a blue repaint 256k (sold for parts)
1995 E320 Polarweiss, 131k (sold for parts)
1995 E300D Polarweiss, 287k (totaled by drunk driver)

If you were viewing the Timing Chain as it was riding on the Camshaft Gear:
Unless the Camshaft Drive Gear is bent I believe that movement would have to be controled by the end play of the Camshaft.

So there should be a spec for the Camshaft end play/thrust.

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84 300D, 82 Volvo 244Gl Diesel

I took a quick look through the Manual and did not see even the Camshaft End Play spec.
Maybe someone has the Tech Manual and the spec will be there.

I also noticed that the Timing Chain Gear is bolted to another Gear on one of the Camshafts; could there be a bolt coming loose?

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84 300D, 82 Volvo 244Gl Diesel