The competition between Heracles and Apollo.
Another famous incident is Heracles so called competition with Apollo, an episode that took place right after the final labour.
It happened so that Heracles desired to take a new wife (remember that the labours was his punishment for killing his first wife). Now Eurytus (not to be mixed up with Eurystheus), prince of Oechalia, had promised that any man who could beat him and his son in an archery competition would win his daughters hand.
Heracles of course won the competition and was to have the daughter as his wife, but Eurytus wouldn’t let someone who had killed his earlier wife and children get married to her. The hero thus left and soon afterwards some cattle was stolen, Eurytes assumed that this was Heracles doing. A man who believed Heracles to be innocent was sent after him, but was killed by the hero (thrown of the walls at Tiryns, se pictures here and here).
Yet again Heracles had committed murder and had to be purified, but this was more difficult than expected. He ended up turning to the oracle at Delphi after being refused a number of times at other places, but even the Pythia (Apollo’s oracle at Delphi) would not do this for him (“[the purification would come] not by oracles” as she supposedly expressed it, Apollod. Bibl. II.6.2). Heracles temper took the upper hand when hearing this and he tried to steal the tripod placed there, “to institute an oracle of his own” (Apollod. Bibl. II.6.2).
This is where the competition came in to the myth, Heracles competing with Apollo for the tripod until Zeus finally separated the two with a thunderbolt. The oracle then told Heracles what to do;
"When they had thus been parted, Hercules received an oracle, which declared that the remedy for his disease was for him to be sold, and to serve for three years, and to pay compensation for the murder to Eurytus."
Apollod. Bibl. II.6.2, translation by J. G. Frazer