The definitive (and recent!) work on this topic, from the asymptotic complexity point of view (which I emphasized in my comment) is due to
Esther Ezra, a student of Micha Sharir.  See especially the paper from her Ph.D. thesis,
"On the Union of Cylinders in Three Dimensions,"
*Discrete & Computational Geometry*, 
Volume 45, Issue 1, January 2011,
Pages 45-64 ([ACM link][1]; [PDF download link][2]).
From the Abstract:

> We show that the combinatorial complexity of the union of $n$ infinite cylinders in $\mathbb{R}^3$, having arbitrary radii, is $O(n^{2+\epsilon})$, for any $\epsilon>0$; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir ...

Deeper into the paper:

> We note that it is crucial to assume that the cylinders are
infinite. Otherwise, the combinatorial complexity of their
union is $\Omega(n^3)$ in the worst case. Indeed, suppose we have
a set of $n$ cylinders, each of which with a sufficiently large
radius and height that is arbitrarily close to $0$. We can now
arrange these cylinders in a (three-dimensional) grid-like
structure, resulting in $\Omega(n^3)$
holes in the union; see Figure 1(a).

<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![Fig. 1a][3]<br />
The general theorem I had in mind in my (hasty) comment is that, $n$ algebraic surface patches in $\mathbb{R}^d$ define an arrangement of combinatorial complexity of $O(n^d)$, where the constant
of proportionality depends on $d$ and the maximum degree of the algebraic surfaces and of the
polynomials defining their boundaries.
This can be found on p.533 of *[The Handbook of Discrete and Computational Geometry][4]*, Theorem 24.1.4, in a chapter by Dan Halperin.

I believe a closed, end-capped, finite cylinder can be partitioned into four surface patches satisfying the preconditions of this theorem.  So we should have $\Omega(n^3)$ from the coins
example above,
and $O(n^3)$ from this general theorem, and so $\Theta(n^3)$ asymptotically.


  [1]: http://dl.acm.org/citation.cfm?id=1929932
  [2]: http://www.cims.nyu.edu/~esther/Publications/cylinders.pdf
  [3]: http://cs.smith.edu/~orourke/MathOverflow/Cylinders3D.jpg
  [4]: http://cs.smith.edu/~orourke/books/discrete.html