The other answers are excellent, but let me add a few
points.

First, with a historical perspective, all the early
fundamental theorems of calculus were first proved via
methods using infinitesimals, rather than by methods using
epsilon-delta arguments, since those methods did not appear
until the nineteenth century. Calculus proceeded for
centuries on the infinitesimal foundation, and the early
arguments---whatever their level of rigor---are closer to
their modern analogues in nonstandard analysis than to
their modern analogues in epsilon-delta methods. In this
sense, one could reasonably answer your question by
pointing to any of these early fundamental theorems.

To be sure, the epsilon-delta methods arose in part because
mathematicians became unsure of the foundational validity
of infinitesimals. But since nonstandard analysis exactly
provides the missing legitimacy, the original motivation
for adopting epsilon-delta arguments appears to fall away.

Second, while it is true that almost any application of
nonstandard analysis in analysis can be carried out using
standard methods, the converse is also true. That is,
epsilon-delta arguments can often also be translated into
nonstandard analysis. Furthermore, someone raised with
nonstandard analysis in their mathematical childhood would
likely prefer things this way. In this sense, the
preference between the two methods may be a cultural matter
of upbringing.

For example, H. Jerome Keisler wrote an introductory calculus
textbook called Elementary Calculus: an infinitesimal
approach, and
this text was used for many years as the main calculus
textbook at the University of Wisconsin, Madison. I
encourage you to take a look at this interesting text,
which looks at first like an ordinary calculus textbook,
except that in the inside cover, next to the various
formulas for derivatives and integrals, there are also
listed the various rules for manipulating infinitesimals,
which fill the text. Kiesler writes:

This is a calculus textbook at the college Freshman
level based on Abraham Robinson's infinitesimals, which
date from 1960. Robinson's modern infinitesimal approach
puts the intuitive ideas of the founders of the calculus
on a mathematically sound footing, and is easier for
beginners to understand than the more common approach via
limits.

Finally, third, some may take your question to presume that
a central purpose of nonstandard analysis is to provide
applications in analysis. But this is not correct. The
concept of nonstandard models of arithmetic, of analysis
and of set theory arose in mathematical logic and has grown
into an entire field, with hundreds of articles and many
books, with its own problems and questions and methods,
quite divorced from any application of the methods in other
parts of mathematics. For example, the subject of Models
of
Arithmetic
is focused on understanding the nonstandard models of the
first order Peano Axioms, and it makes little sense to
analyze these models using only standard methods.

To mention just a few fascinating classical theorems: every
countable nonstandard model of arithmetic is isomorphic to
a proper initial segment of itself (H. Friedman). Under the
Continuum Hypothesis, every Scott set (a family of sets of
natural numbers closed under Boolean operations, Turing
reducibility and satisfying Konig's lemma) is the
collection of definable sets of natural numbers of some
nonstandard model of arithmetic (D. Scott and others).
There is no nonstandard model of arithmetic for which
either addition or multiplication is computable (S.
Tennenbaum). Nonstandard models of arithmetic were also
used to prove several fascinating independence results over
PA, such as the results on
Goodstein sequences,
as well as the
Paris-Harrington theorem on the
independence over PA of a strong Ramsey theorem. Another
interesting
result
shows that various forms of the pigeon hole principle are
not equivalent over weak base theories; for example, the
weak pigeon-hole principle that there is no bijection of n
to 2n is not provable over the base theory from the weaker
principle that there is no bijection of n with
n^{2}. These proofs all make fundamental use of
nonstandard methods, which it would seem difficult or
impossible to omit or to translate to standard methods.

helpfulnon-standard analysis is. If you wanted to know how unhelpful it is, you would ask for theorems thatcannotbe proved using non-standard methods. $\endgroup$ – François G. Dorais♦ Feb 24 '10 at 22:57