Answer: Sam Rice batted .321 in his 1,714 at bats during his 40s, including .349 at age 40. That year, he had 207 hits and is still the only player in baseball history to do so during his 40s.
He played until he was 44, when het batted .293, the lowest batting average of his career.
In addition to having the highest total batting average, he ranks in the top 10 in many
statistical categories for players in their 40s (for those with 1,240 Plate Appearances), including:
1st in At Bats per Strikeout (36.5)
2nd in Doubles (95), Triples (32) and Runs (327)
3rd in hits (551) and Runs Created (271)
4th in OPS (.801) and Total Bases (722)
5th in games played (543), At Bats, and On Base % (.379)
6th in Slugging (.421)
9th in Stolen Bases (31)
Ty Cobb has the highest single season batting average for someone in their 40s (.357 when he was 40 in 1927), but he only played until he was 41, so he doesn't qualify per Baseball Reference standards requiring 1,240 plate appearances.
A few other tidbits about Sam Rice:
1) Before starting his baseball career he had married, had two daughters, and then lost his entire family when a devastating tornado swept through his hometown of Morocco, IN in April of 1912 (he was away trying to win a job with a team in Galesburg, IL at the time).
2) Bereft of relatives, he drifted around the country for a couple of years, joined the Navy, then was discharged in 1915 to play for the
Petersburg Goobers of the Virginia League, where he both pitched and played the outfield. The team's owner had a $600 debt to Senators owner Clark Griffith, and unable to repay it, gave him his best player as compensation. When he arrived in Washington, Griffith could not remember his first name - Edgar - and told reporters he had obtained "Samuel Rice". The name "Sam" then stuck for the rest of his life.
3) Rice was the first player to hit an inside the park home run at Yankee Stadium. He had
21 career inside the park home runs,
9 of which occurred at Griffith Stadium in Washington, DC. He never hit a single home run over the fence at Griffith Stadium.
4) Joe Judge and Sam Rice played together as Nationals for 18 consecutive seasons, a teammate longevity record they held until it was tied by George Brett and Frank White of the Kansas City Royals in 1990, and passed by Allen Trammel and Lou Whitaker of the Detroit Tigers in 1995.
5) After baseball, Rice returned to the simple life; he was the owner of a chicken farm near Olney, Maryland. During World War II, he employed Japanese-Americans who were relocated from their homes on the West Coast due to the unfortunate government policies at the time.
6) Rice stayed close with his teammates after baseball, many of whom bought farms near Washington DC after retiring. They frequently got together, but as time went on, many of those get together were funerals, including those of Eddie Foster in 1937, Ray Morgan in 1940, Walter Johnson in 1946, Clark Griffith in 1955, Jim Shaw, the pitcher who Rice replaced on the mound in his major league debut, in 1962, Joe Judge in 1963, and Nick Altrock in 1965.