Fourier series is to Fourier transform what Laurent series is to ...?

In the same way that there is a discrete $\rightarrow$ continuous transition as one goes from Fourier series to transforms, one can make a continuous analogue to Taylor series in the form of the Laplace transform. This is beautifully explained by Arthur Mattuck in this video and this one, which I heartily recommend, but the gist is this:

Start with a Taylor series of the form $\sum_{n=0}^\infty a_nx^n=:A(x)$, and change the discrete variable $n=0,1,2,\ldots$ for a continuous variable $t\in(0,\infty)$, to get $\int_0^\infty a(t) x^t dt$. From this it is only a cosmetic change to define the variable $s=-\ln(x)$, which is positive and real in the interesting range for $x$, $0<x<1$, and this makes the integral look like $$\int_0^\infty a(t)e^{-st}dt=:A(s),$$ which is just a standard Laplace transform. Typically $A(s)$ will be defined in a half-plane $\text{Re}(s)>s_0$, as long as $a(t)$ is of exponential class for $t\rightarrow\infty$.

If you now want a continuous analogue of Laurent series, then all you need to do is extend your integration to minus infinity as well: $$\int_{-\infty}^\infty a(t)e^{-st}dt=:A(s).$$ This will make the typical domain for $s$ a strip of the form $s_0<\text{Re}(s)<s_1$. This corresponds to $x=e^{-s}$ being in an annulus around zero, which is also the typical domain of convergence of Laurent series.

You are also much more constrained in your choice of $a(t)$, in the same way that coefficient sequences are more restricted for Laurent series. This is in a way more serious here, because the class of 'interesting' functions which are now available is indeed noticeably smaller than the equivalent restriction in Laurent series. (In particular, things like 'folding' a sequence so that e.g. $a_n=1/(|n|!)$ become more problematic, and can cause $a(t)$ to not be analytic anymore.) This means that the transform is slightly less useful, and indeed it is much less common in the literature than the standard Laplace transform.

You may want to look into the Mellin transform. The basic idea is similar to what you seem to have in mind, but the integration contours are different from what you guessed. Incidentally, one side of the Mellin transform applies even outside the context of analytic functions. Of course, the Fourier, Laplace and Mellin transforms are fundamentally related, through changes of integration variables.