Only twice have I had difficulty getting debris out of the tyre.
1. An ultra-thin thorn. So fine it was impossible to get a hold of it without snapping it off. I poked a screwdriver into the hole and mashed the residue to a soft pulp. It wasn't any more trouble.
2. A hedge trimming. It was about 4mm diameter and jammed too tight to pull it, so I trimmed it flush with the tread using a penknife, then pushed it through with the screwdriver. The damage caused the bracing in the tyre to fail a few hundred miles later though.
I've had assorted metalwork too, including a screw that damned near punctured the bed of the rim as well, but nothing I couldn't remove.