Though dice are typically used to randomize events or check against the odds of success, they also have a good use in keeping records and are probably the cheapest way to do it.
There are many non-pencil and paper games that want you to keep track of hit points during game play. This is most easily done using dice near the targets, since writing them down means you have to keep erasing and remarking. Dice are also useful when keeping track of ammunition. In cases like this, I highly recommend using a very specialized color of dice to make it clear that its purpose is for record keeping and that it should not be rolled. One clever player I knew used bright pink dice for this purpose. He never mistook it for anything but its proper use.
There’s a game right now that uses d4s as pyramids. The size of the pyramid is based on which number is on the top. This beats a game that has mere plastic pyramid pieces because its size can change based on the number. This is a clever way to use dice because it provides alterations without requiring multiple pieces (which raises the price of the game.)
I use dice in my game, Command Combat: Civil War, to denote exhaustion and morale as well as the amount of fire coming in on them. You place a d10 in front of a unit to show how much firepower is being shot at them, and another dice behind them to denote their fatigue and morale. The player simply rolls against those numbers to see if the units are hit and if they run. By doing this, their morale and fatigue changes throughout the game and makes it possible for units to become more and less exhausted throughout the course of the battle.
One of the very clever uses of dice for marking rather than rolling is in the miniatures game Bolt Action. Each player gets a number of specialty dice based on the number of units they have. These specialty dice are all thrown into a bag and shaken. Then one is pulled out each turn. That denotes which side is being activated. Each side has an order, and the player takes the die and chooses one side of it. The die is placed on the table with that order facing up. The unit then performs that order. This makes a little bit of randomization using the die, but not entirely. I think what would be fun in a game like this would be to have a chaos army that never gets to choose its order but instead has to roll to see what action it performs.