New York Tribune (July 28, 1873) (The Maumee River is in northwestern Ohio and northeastern Indiana.):

"Even here on the Maumee...fever and ague and small potatoes were set to music, living long...in the immortality of indifferent song:

The potatoes they grow small,
    On Maumee, on Maumee,
And they eat them tops and all
    In Maumee."

Detroit Free Press (Feb. 6, 1881):

"Sixty years ago the home of Congressman Hurd was only heard of in song as the place 'where potatoes they grew small on Maumee.'"

                                 *

St. Louis Post-Dispatch (Dec. 19, 1897), p. 14:

Potatoes they grow small
   In Kansas;
Potatoes they grow small
   In Kansas;
Potatoes they grow small,
And they dig ‘em in the fall,
   And they eat ‘em, skin and all
   In Kansas.

“He also harped a stanza about roosters which lay eggs in Kansas.”

                                 *

Chicago Daily Tribune (March 19, 1928):

"The late [vaudevillian] Eddie Foy [1856-1928] sang:

Potatoes they grow small in Kansas,
Potatoes they grow small in Kansas.
   They plant them in the fall,
   And it doesn't rain at all,
And they eat them skins and all in Kansas.

There are no old maids in Kansas.
There are no old maids in Kansas.
   When a girl is thirty-one,
   A policeman gets a gun
And shoots her just for fun in Kansas.

"And repeated calls for encores would exhaust Eddie's supply of verses
until he simply sang the air to the words of 'Rub-a-dub-dub, rub-a-dub-dub.'"

                               *

There is also a scurrilous anti-Yale version:

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