I presume it should read 'My horse ran the race of his life'. The * of one's life is the best *, most satisfying *, most challenging *, most enjoyable*, etc. To have the time of one's life is to have a very good time indeed.
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Mister MicawberYou can replace the asterisk with lots of nouns of strenuous activity.What about enjoyable activities? I heard a sentence describing a dog's excitement being washed in a bathroom - "He's having the time of his life."