Exercises to do while alone

• Figure out your goals, and then make plans for when you get off work to optimize for those. Working as a cashier doesn't seem optimal for almost any purpose--maybe you could start by figuring out how to make money more efficiently, if that's your goal?
• Learn the major system or memory palace. This would let you store a list of things to think about or do when at work. It's also quite easy to practice while at work, once you get the basics down. I'd recommend this first, if you really won't be allowed to write.
• Solve problems. See what problem-solving methods work and which don't. See what kinds of problems you are worst/best at, and become better at those. Math problems, world-modeling (prediction and underlying event deduction), and introspection are especially easy to do in your head.
• Try to figure out why stuff around you is the way it is. (Why did that person buy that item?). Make predictions. Calibrate and get higher accuracy as well.
• Introspect. Find out why you believe what you believe, and whether you should.
• Don't improve your rationality, do something else with your time.
• Optimize your job as a cashier, as much as is possible. Figure out how to do stuff in the least time. Experiment when interacting with customers to see if you can get tips or interesting conversation. Get a different job (manager?) at the same establishment somehow. A useful problem will motivate you more than a non-useful problem.
• Combine all these.

Do you have access to a smartphone (and permission to use it)? There are many implementations of dual n-back for the popular mobile operating systems. (I can't try any of those applications, so I cannot offer any advice on which ones are good/bad)

ETA: gwern's FAQ explores dual n-back well (and some lists some other mental exercises towards the bottom).

Do you specifically mean "nobody is at the register" or "greeting people at the register counts as brain-down-time?"

Because it's worth noting that whenever there is a person at the register, you have a valuable opportunity to improve your social skills.

I'm in a similar boat; I work overnight at a well-known US gym for 8 hours, and the shift is so slow I'm allowed to pull out my laptop, books, phone, whatever I want. (No internet, though, except on my phone.)

That said, I recommend worldbuilding. I do it for my tabletop games, but you could just as easily do it in the modality of a novel. Of particular use would be the creation of the histories of countries and political systems; in this way, you could experiment with social conventions... perhaps some outside your own culture. You are, of course limited by your imagination... but only your imagination.

You could try doing the exercise I posted on here.. It's more fun when going around a circle, but it might be a nice way to pass the time and work on your idea-generating muscles.

As an add: I spend quite some time in the steam room, which has the added property of not being friendly to technical devices. What I do so far is to prep my memory system, and to think about single issues for a while. But I am happy to collect more suggestions, maybe a few from here are suitable for that too like mental arithmetics.

Why not practise mental arithmetic, like 45 times 23. It's not really rationality, but can't hurt. It's probably good for your brain somehow.

Or you could try doing fun pointless economics or physics calculations. If you're a cashier at a supermarket, you could calculate how far the chemical potential energy in a can of soup or whatever would propel it into the air, and do the calculation for as many products as you can find. Or figure out what proportion of the money that comes through you would have had to have stolen and invested twenty years ago on order to get your current salary. Or something like that. I dunno.

(Note: Here in Australia, cashier might have a different meaning. I hope I didn't offend you by implying you were a check out guy in a supermarket.)