In Calc 1, you spend time learning how to take derivatives and integrals of ln(x), e^x, and trig. Then in Calc 2, you're told how to convert these functions into Polynomial functions via Taylor/Mac. Motivation is that Poly is easier to deal with. But, you just learned how to take deriv/int of those in Calc 1.
Anyone got a case or situation where you can't work with the function, so you convert to Poly? This is a real question, so please don't ban it.
On
Note the Taylor and Maclaurin polynomials are only approximations. The Taylor and Maclaurin series are power series, not polynomials (unless the function itself is a polynomial).
To answer your question: It may not be possible to write your function down explicitly in terms of elementary functions, so you may have to settle for a power series representation. One function that comes to mind is the Lambert $W$ function. This is an inverse function for $f(x) = xe^x$. In other words, $W(x)e^{W(x)} = x$. It can be represented as $$W(x) = \sum_{n= 0}^{\infty} \frac{(-1)^{n-1}}{n!} n^{n-1} x^n$$
The point of Taylor polynomials is to approximate a function. The simplest case is just to compute numerical values of the function. What is the value of $\ln1.3$? $\sin.32$? $e^{4.1}$? With Taylor polynomials, you can easily compute these to any desired accuracy. Of course, these days, you can punch these expressions into your calculator, but there are many, many other cases where Taylor polynomials are useful.
One example is in computing complicated limits. There are many times when approximating a function by a polynomial makes it easy to compute a limit that would otherwise be daunting. Another example is numerically computing definite integrals. For example, the error function is of great practical importance, and it's vital to know its values, but $e^{-t^2}$ doesn't have an elementary integral. Enter the Taylor polynomial.
It would be hard to overstate the importance of Taylor series, both in theory and applications.