For every real number, you can create rational numbers that are arbitrarily close to it. As a result, rational numbers are a sufficient field for every real-world problem involving real numbers, and as a result of that computers are useful for scientific purposes.
Is something similar true of algebraic functions? When I conduct an experiment I can constrain the values of a function to an arbitrary but finite precision. Will I ever be able to rule out the theory that the function I am measuring is algebraic, or are there algebraic functions arbitrarily close to every other function? (I'm excluding, of course, pathological functions like the ones that are continuous nowhere.)
If "close" means uniformly close on the whole real line, then no. For example, any algebraic function $f$ with $|f(x) - \sin(x)| < 1$ for all real $x$ would have infinitely many real zeros (by the Intermediate Value Theorem), but there are no such algebraic functions except $0$.