The Chicago Colts set the record for most runs in a game on June 29, 1897 when they scored 36 against the Louisville Colonels.
Since 1900, two teams hold the record. The Boston Red Sox scored 29 against the St. Louis Browns on June 8, 1950. The Chicago White Sox tied the record five years later when they defeated the Kansas City Athletics 29-6 on April 23, 1955.
The way runs are being scored today, we might not have to wait very long to see this record broken.