First off, lets look at an abstract game that doesn't require anything more than the printed board and a pencil or pen. Hex.
Hex was independently invented by two different mathematicians in the 1940s, Piet Hein and John Nash (yeah, the Beautiful Mind guy) It has one of the simplest rule sets you're ever going to find for boardgame. You play on a hexagonal grid, placing stones or drawing your mark on the individual hexes. Each player is assigned to parallel sides of the board and the goal is to connect your sides.
In other words, take the basic idea of Tic-Tac-Foe and scale it up by a thousand.
A few interesting facts about Hex. One, while 11 x 11 seems to have become the standard board size, there's no restriction on how big or how small the board can be. Two, it has been conclusively proven that it is impossible for there never to be a draw in Hex. Third, Hex popularized the pie rule. That's when the second player on their first turn can either make a move or switch colors.
Out of the three games that I'm going to talk about, Hex is oldest and really the best. However, it is also a very unforgiving abstract. The better player will always win unless someone is hitting them over the head with a hammer.
Quite frankly, if you don't like abstracts and you want at least a little bit of luck in your games, then you're better off giving Hex is a miss.
The next game that I want to recommend is Reiner Knizia'' Decathlon. Yes, I don't seem to be able to write about or games without mentioning Reiner Knizia. It will happen someday though, I promise.
The Decathlon is a series of little dice game, each one for a different event. While they all can be compared to Yahtzee, each one has its own twists and has at least a little to do with the actual athletic event.
Different folks have formatted in ways over the years. Still, all you need is a print out or two of the rules and scoring sheets, plus eight dice and a pencil to keep track of the score.
It's not Knizia's best game but it is his best free game and a fun little dice chucked. If you're even a little interested, it's super easy to try out and you can even play it solitaire.
The last game I want to mention is Pagoda, which is an abstract that uses dice for the pawns, with a little dice rolling thrown in for good measure. It was originally a published game but the publishers have now released it as a free game that you can make yourself.
You play Pagoda on a circular board that shows the bird's eye view of a mountain with a pagoda on the top. Your goal is to have your five dice trace of line from the bottom of the mountain up to the top and back down again.
What makes the game interesting is that the pips on the dice determine if the dice is a worm, a goat, a human or Buddha. You start off with one worm, two goats and two humans. However, every time a pawn dies, you reroll it to reincarnate it.
(The six is the Buddha, by the way, who you can only achieve by reincarnation. The Buddha is not placed on the board but is considered to be on any one space. So, if you somehow ended up with five sixes, you would automatically win.)
While I would not consider Pagoda to be as strong as either Hex or Reiner Knizia's Decathlon, since it can end up bogged down, I do really like the reincarnation mechanic. It gives the game very unique feel. Plus, it's pretty good for a game that you just have to print out one page for the board and add ten dice.
As I mentioned before, print and play is very much a niche in the gaming hobby. Some projects can end up being quite a bit of work and many projects can end up not being worth playing. However, there is enough out there that can be fun, even without a lot of work.