I know of two other games that were widely played. The first is chunkey and the people best known to play it would be the Mississippians. Chunkey consists of a small stone disc, sometimes concave on both sides, which was rolled onto a large flat playing space. Two teams of players would chase after the stone and try to knock it down with long thin sticks. Betting became so problematic for some people that they sold themselves into slavery.
The other is the Mesoamerican ballgame (Ōllamaliztl in Nahuatl or pitz in Maya) which is famous for its often I-shaped ball courts. The rules between peoples, cultures, and time periods is still poorly understood. However a modern variant is played called ulama.
Plains tribes had a few different games:
Shinny- Similar to field hockey, the object was to hit the opponent's goal post with a ball struck by a curved stick. Common for the Northern Cheyenne and the Crow.
Ring the Stick- A ring is attached to a stick with string. The goal was to toss the ring up and catch it with the stick. Widely played by many tribes as a way to work on hand-eye coordination.
Doubleball- Similar to the popular BBQ game Ladderball, the object was to wrap a pair of balls connected by string around a horizontal bar by tossing them. Multiple bars with varying point values were used. Popular with the Chippewa and Cree.
Wrestling and foot races were also popular among the Native Americans, that were forced to live in the Indian Territory. Jim Thorpe wrestled and competed in sprint races and longer distance races as a young boy, in the 1890s.
Source: Robert Wheeler "Jim Thorpe: The World's Greatest Athlete" (1979)
Back in August, I made a post in similar topic that covers lacrosse and a few other popular games (including chunkey, which /u/Mictlantecuhtli already mentioned). I thought I might take this opportunity to discuss a few more that I didn't mention last time:
#The MOCCASIN GAME
Related to the Hidden Ball Game of the Southwest, Moccassin is a game of deception and distraction. One player hides a ball (in the historic period, this was usually a musket ball) under one of 4-6 moccasins laid out on a playing mat. In the Ojibwe version of the game, it's almost always four moccasins (or faux-moccasins made specifically for the game) and instead of one ball, it's four - one of which is distinct from the other three. The other player, who watched as the ball was hidden, attempts to find it again, but not right away. Ideally, he needs to eliminate the places where the ball is not first. Finding it too soon awards his opponent points. Two of the non-hiding spots need to be eliminated first before the seeker can start earning points for himself. Eliminating all the non-hiding spots before revealing the ball allows the seeker to confiscate points from his opponent. This game was adopted by European colonists, though they used their hats instead of their shoes. On a similar note, the variants in the Southwest tend to use tubes and cups instead of moccasins.
#RING AND PIN
Though Culin calls this game, generically, "ring and pin," he admits that rings are actually rarely used in it. It's similar to the European Ball and Cup game. It had a wide distributions, but its primarily played in northerly communities, and those living in the Arctic have some of the most elaborate variants of the game. Down here in the Eastern Woodlands, the most common variant involves a string of 6-8 deer phalanges. At the other end of the string is a long pin. The player holds this pin and tosses the bones into the air, attempt to catch the bones as they fall. Each bone (or sometimes a set of bones) will have a number of points assigned to them, with the bones closest to the pin being worth the most points. Another version, popular in the Wabanaki Confederacy, used a circular target of moose hide which had a pattern of holes in it, with the objective being to get the pin through certain holes to score the most points. The Wendat version was one of the simpler variants. It used a corn husk ball, which had to be caught on the pin, but didn't have any scoring method beyond "caught or not caught" as far as we know today.
#BOARD GAMES
Culin lumps all these all into his category of games influenced by or adopted from Europeans, though the ethnohistorical sources indicate that several of these games had "been long known" to their Native players and were played with rules that were unfamiliar to Europeans. If they were picked up from Europeans, these games quickly evolved into new creatures. I'll mention two board games developed by Eastern Woodland cultures here.
Musinaykahwhanmetowaywin is an asymmetrical game played by the Ojibwe and Cree. The board is a cross-shaped pattern of five squares, each of which is crisscrossed by four lines (vertically, horizontally, and diagonally). At each intersection is a slot of one of the fourteen peg-pieces. Thirteen of these pieces are small human-shaped pawns; the last is a much larger ogimaa piece (to update the spelling used in Culin's book. Unfortunately, I don't know enough Ojibwe to even begin to figure out what the game's name should be in modern orthography). One player controls the pawns, while the other controls the ogimaa. On it's turn the ogimaa can move in a straight line as it pleases, capturing pawns as it goes (though there does seem to be a way for the pawn-player to protect his pieces, the specifics there aren't mentioned). The pawns can't capture the ogimaa directly. Instead, they have to herd it into a position where it can't move away,
Ko-ko-nag'n is a Passamaquoddy board game that has been compared to checkers (a game that was adopted from Europeans and widely popular among the members of the Wabanaki Confederacy). We don't know much about this game except that it was regarded as particularly challenging (Euroamerican writers never seemed to have picked up the rules and only a few Native players were said to have been any good at it), circular and square pieces of differing values were employed, and unlike most games, men and women could play against one another.