It’s been a while since I had located a good photo for sleuthing, but this is definitely a good one.
When the now-iconic center-field scoreboard was built atop the new outfield bleachers at Wrigley Field in 1937, it had no clock on the top, as shown in this rare color photo from 1941:
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You can also see that the original color was brown. The board wasn’t painted in its current olive-drab green color until 1944.
The color photo above was taken May 25, 1941, as sleuthed in the article link above.