My favorite was at the Falling Leaf Rally many years ago. An R90/6 or similar vintage Airhead refused to start on a frosty Sunday morning. We tried jumper cables. Then we tried pushing. At pushing speed the bike went into a death-wobble and was absolutely uncontrollable. We tried several pushers and several brave riders all to no avail. We finally got it started using jumper cables from a large diesel truck, but not by pushing it.
Once back to Kansas the owner brought it to my house. I found two problems. The starter problem was a binding starter due to a worn nose bearing. The wobble was caused by very loose steering head bearings.
I asked the owner how he could even ride the thing. He said that starting out it would wobble but he would just add power/speed and it was OK. Coming to a stop, he said, was a bit dicey so he would just grab more brake to stop more quickly.