I think the motivation for the Hirzebruch-Jung algorithm is not the algorithm itself, but the fact the it yields *directly* a very interesting continued fraction expansion. It is those, now called **Hirzebruch-Jung continued fractions**, that have a wide number of applications. A bit more on that later.

Using your notation (but changing $b$ with $r_o$ for clarity).

$$ \begin{array}{rcl}
a &= & e_1r_0 -r_1\\
r_0 &= & e_2r_1 - r_2\\
r_1 &= &e_3r_2 -r_3\\
&...\\
r_{k-2} &= &e_ir_{k-1} = e_i
\end{array}$$

By simple division you get:

$$ \begin{array}{rcl}
a/r_0 &= & e_1 -r_1/r_0\\
r_0/r_1 &= & e_2 - r_2/r_1\\
r_1/r_2 &= &e_3 -r_3/r_2\\
&...\\
r_{k-2}/r_{k-2} &= &e_i
\end{array}$$

And from that you can simply read off:

$$a/r_0 = e_1 -\frac{1}{e_1-\frac{1}{...-\frac{1}{e_i}}}$$

It is those finite continued fractions $a/r_0$ that have, among many others, applications to toric varieties. To quote from D. I. Dais' "Geometric Combinatorics in the Study of Compact Toric Surfaces" (2000):

"Examining two-dimensional toric singularities "under the microscope"
one discovers a peculiar algebro-geometric world endowed with a rich
combinatorial structure. Viewed historically, everything begins with
Hirzebruch-Jung continued fractions."

For some more motivation and applications, I quote form a very recent (2015) paper:

"Hirzebruch-Jung (H-J) continued fractions are widely used in various
branches of mathematics as well as in theoretical physics. First of
all, HJ-continued fractions arise naturally in the minimal resolution
of cyclic quotient (that is, Hirzebruch-Jung) surface singularities of
the type $\mathbb{C}/\mathbb{Z}_p$, which is also known as
HJ-resolution" [...]

HJ-continued fractions are used also to describe the plumbing
decomposition of the other type of link of surface singularities,
namely, Seifert fibered homology spheres (Sfh-spheres), particularly
Brieskorn homology spheres (Bh-spheres) [...]

In condensed matter theory, the HJ-continued fractions are used to
describe fractional quantum Hall (FQH) systems with k levels of
hierarchy [...]

Also, in topological string theory, the structure of internal space
(Calabi-Yau threefold) can be encoded in terms of HJ-continued
fraction expansion of positive integers, which are the topological
invariants of the internal space and define the mode of interactions
between D-branes"