Yes, there are many such fields.

(Edit:  Let me put up here that "many" is still finite, and only in the hundreds if my list below represents most of the known examples.  Note that "state-of-the-art" is still pretty primitive here -- infinitely many examples would give infinitely many number fields with class number one, which we don't know to exist (but strongly strongly suspect to!).)

As Torsten brings up (and what is behind the exercise in Washington that dke mentions), the Odlyzko bounds are really the key here.  Briefly, the remarkable result that comes from these bounds is:

<blockquote>
The ring of integers in any number field with sufficiently small root discriminant has trivial fundamental group, i.e., admits no non-trivial unramified extensions.
</blockquote>

The "sufficiently small" is large enough, with the Odlyzko bounds, to generate fairly large examples of what you're looking for:
<ul>
<li>  All quadratic imaginary number fields with class number 1 (of which there are 9)
<li>  All cyclotomic fields with class number 1 (of which there are 30)
<li>  All real quadratic fields with prime-power conductor $\leq 67$ (I count 9).
<li>  The top of the class field tower over a quadratic imaginary number field $K$ for many many $K$, including at least all such number fields with conductor $\leq 1000$ (a couple hundred).
</ul>

The last two of these are work of Yamamura (results spread out, but see "Maximal Unramified Extensions of Imaginary Quadratic Fields of Small Conductors", parts I and II).

While I'm here, this last one answers affirmatively the question brought up by Minhyong in the comments.  For example, $\mathbb{Q}(\sqrt{-771})$ has a ring of integers with fundamental group $S_4\times \mathbb{Z}/3\mathbb{Z}$, for this is precisely the Galois group of the Hilbert class field tower over this base.  Finite but non-trivial.  Many other examples of fundamental groups that can show up in this way are found in Yamamura.

I think the natural follow-up (and incredibly interesting) question is to classify exactly which groups can show up in this fashion.  This is completely intractable at the moment, modulo some easy partial results in both directions.  Edit:  For example, by class group considerations, it is impossible to have a quadratic imaginary number field with etale fundamental group isomorphic to $\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$.