I know the question is vague, but let me give some background for it:
I am part of a renaissance faire guild. Specifically a travelling tavern that is mostly a support guild (we give a place for other guilds to come and convene and give an idea of what a tavern in the time was like - mead and cider is served but not to the public).
My role in the guild is the games master. I don't do this as much for the other guilds because they come to relax, but I do enjoy teaching patrons about various games. The problem is that there aren't many games I can teach them that aren't already around or that I know too much about.
I do enjoy "liars dice" (similar to the card game "bullshit" and is seen in one of the Pirates of the Carribean movies, played amongst the captains) although I don't know if that is the right thing to call it, or "shove a penny" which is similar to shuffleboard but played on a tabletop with coins (slide the coin into a scoring region but not off the table).
I would like to get into something else, specifically something that would be a betting style game and something easy and quick to teach because the public has a short attention span. I have ideas for things, but nothing that is nearly accurate and I've not been given permission (by the guild leader or my wife) to expand into games that I can't show are somewhat accurate for the time.
I've tried looking it up and I can't find much on it because most games lasted longer because drinking and relaxing wasn't as much of a "one drink and go" type of thing. Can anybody here PLEASE give me some insight into what types of games were played or where I can find information on it?
Tragically, my favorite historical drunk game just isn't going to work here. Late medieval students at the University of Paris used to get wasted, break into the lecture halls...and play tennis.
That said, it is frustrating to always read that people played "cards and dice" and never get any details. When one is described, it's usually shove-halfpenny or shove a penny, which you already mentioned.
A tavern version of skittles--outdoor bowling--is, however, attested from 1626 (that's when it shows up in court records as the source of a barfight, suggested it was played earlier). Indoors, this would involve setting up some kind of pins or cups--skittles--on a table and rolling a ball into it. The easy version is how many skittles can you knock over, but variations included trying to knock over skittles in certain positions but not others, and in predicting how many skittles your or your opponent's roll would knock over. This is easy to play and easy to wager on. For thise of you who haven't researched this topic like OP--early modern bar bets were typically for a round of drinks, maaaaybe a penny. While amounts like these could add up over time, the nature of clerical vitriol against gambling and the relative laxity of civic ordinances in the early modern period, generally speaking, suggests that gigantic ruinous individual wagers were rare.
Andrew Leibs' book Sports and Games of the Renaissance, published by our "daily life" experts Greenwood Press, actually contains the rules for a fair number of card games attested in the era, like maw from 16C Ireland. I'd definitely suggest picking up a copy of this or fighting with the Google Books preview for access to the pages around, oh, 90 and following.
The other type of game that sometimes spilled over into tavern culture is what is more commonly known as parlor games. These are games of wit rather than luck/skill, so they acquired entire new dimensions when played by drunk people. Parlor games are as diverse as the times they were played, and at least some of the culture around them involved coming up with new ones on the spot.
These often bear a resemblance to new-group icebreaker activities. Go around the circle and each person must contradict what the person beofre them said while still carrying on a conversation. Make up a motto and mascot for the next person in line, and try to outdo the others in fittingness (or in crudeness). In terms of primary sources, Baldesar Castiglione's Book of the Courtier is probably the most accessible in English translation. Obviously from the title, Castiglione discusses hese as the makings of polite society. Nevertheless, the lurking bawdy underside of them is evident in basically all the games, along with the insistence (though it is stronger in passages from other authors on parlor games) on the need for sobriety of drink and mind while playing.
Sadly, George McClure's book that is literally called "Parlour Games and the Public Life of Women in Renaissance Italy" is almost entirely unconcerned with the games themselves. Come on, man. Scholars are people, too. Give the people what we want.