You seem to be asking for examples of random variables realized concretely as measurable functions on a probability space. This runs rather counter to the usual point of view of probability theory, which only cares about the distributions (more generally, joint distributions) of random variables. (In particular, as Piotr points out, it's relatively elementary to construct a sequence of independent random variables with arbitrary distributions, defined on [0,1].)

That comment notwithstanding, here are some examples:

Put the uniform probability measure on {0,1}^n, or more generally the n-fold product of a measure which puts mass p on 1 and mass (1-p) on 0. The function $f(x) = \sum x_i$ is a binomial random variable.

Put the rotation-invariant probability measure on the sphere $\sqrt{n}S^{n-1}$. The function $f(x) = x_1$ is a random variable which converges in distribution to the standard Gaussian distribution when $n\to \infty$. (This is sometimes called the "Poincare limit" because it was first observed by Maxwell and first rigorously proved by Borel.)

Put the uniform probability measure on the permutation group S_n. The number of fixed points of a permutation is a random variable which converges in distribution to a Poisson distribution with mean 1. (Showing that the mean of this random variable is 1 for every n is a nice exercise in the method of indicators.)