A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock."

There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational.

Anyway, this started me wondering about how to improve the design of this clock from a mathematical standpoint. Here's my formulation of the problem:

Find 12 numbers that have been proved to be irrational to place around the outside of a clock.

Each of eleven of the numbers must approximate as closely as possible one of the integers from 1 through 11. The 12th can either be just smaller than 12 or just larger than 0.

The numbers must have expressions that are as simple as possible (in the spirit of - or even more simple than - those in the clock given in the picture here). Thus, for example, no infinite sums, no infinite products, and no continued fractions. Famous constants and transcendental functions evaluated at small integers encouraged.

Expressions should be as varied as possible. Better answers would include at least one use of trig functions, logarithms, roots, and famous constants.

Obviously, goals 2, 3, and 4 act against each other. And, as Jonas Meyer points out, "as closely as possible" and "as simple as possible" are not well-defined. That is intentional. I am afraid that if I tried to define those precisely I would preclude some answers that I might otherwise consider good. Thus, in addition to the mathematics, there's a sizable artistic component that goes into what would be considered a good answer. Hence the "soft-question" tag. I'm really curious as to what the math.SE community comes up with and then what it judges (via upvoting) to be the best answers, subject to these not-entirely-well-defined constraints.

Note that the designer of the clock given here was not trying to approximate the integers on a clock as closely as possible.

Finally, it's currently New Year's Day in my time zone. Perhaps a time-related question is appropriate. :)

Note: There is now a community wiki answer that people can edit if they just want to add a few suggestions rather than all twelve.