Why adjoining non-Archimedean element doesn't work as calculus foundation? Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used in non-standard analysis. Why does this approach fail?
 A: The construction you link to will work as far as it goes, if your only goal is to produce some non-Archimedian ordered field. However, the model-theoretic construction is much stronger; it will guarantee that every formal statement that is true about the standard reals is also true for the non-standard reals, as long as you replace all constants in the statement with their non-standard counterparts. Without this, it could be difficult to transfer results between the standard and the non-standard worlds.
For example, if one takes your simpler construction and adjoins a square root of $-1$, will the resulting field satisfy the Fundamental Theorem of Algebra? You would have to repeat the entire proof from the real case, if it goes through at all. With the heavy-caliber nonstandard analysis, you can just says that every nonconstant degree-$n$ polynomial over $\mathbb C$ has a root in $\mathbb C$, thus every nonconstant degree-$n$ polynomial over $\mathbb C^*$ has a root in $\mathbb C^*$. Case closed.
A: If the assumption of the question is that using a simple-minded non-Archimedean field cannot be used to get interesting mathematical results, that assumption can be challenged.  For example, Dehn used a simple-minded non-Archimedean field to resolve one of Hilbert's famous problems; see here.
A: To perform nontrivial analysis in a nonstandard extension of $\:\mathbb R\:$ requires much more than simply the existence of infinitesimals. One needs some efficient general way to transfer (first-order) properties from $\:\mathbb R\:$ to the nonstandard extension. This is achieved by a powerful transfer principle in NSA. Lacking such, and other essential properties such as saturation, one faces huge obstacles.
One need only examine earlier approaches to see examples of such problems. For example, see the discussion of the pitfalls of Schmieden and Laugwitz's
calculus of infinitesimals in Dauben's biography of Abe Robinson
(lookup "Laugwitz" in the index) and, for much further detail, see D. Spalt: Curt Schmieden’s Non-standard Analysis, 2001. In short, viewed in terms of
ultrapowers $\mathbb R^\mathbb N,\:$ the S&L approach mods out only by a Frechet filter on $\mathbb N\:$ instead of a free ultrafilter. Thus one loses full transfer of first-order properties, e.g. one obtains only a partially ordered ring, with zero-divisors (i.e. there are $r,s\neq 0\,$ with $rs = 0)$. Without all of the essential properties of $\mathbb R,\:$ and without a general transfer principle, one obtains a much weaker and much more cumbersome theory as compared to  Robinson's NSA.
Another point which deserves emphasis is the role played by logic, in particular the concept of formal languages. One of the major problems with early approaches to infinitesimals is that they lacked rigorous model theoretic techniques. For example, without the notion of a (first-order) formal language
it is impossible to rigorously state what properties
of reals transfer to hyperreals. This logical inadequacy is one of the primary sources of contradictions in the earlier approaches.
Abraham Robinson wrote much on these topics. It is said that he knew more about
Leibniz than anyone. His lofty goal was to vindicate Leibniz'
intuition, and to reverse the historical injustices done to
him by many Whiggish historians. See Abby's collected papers
for much more, and see also Dauben's superb biography of Robinson.
For a brief introduction to the ultraproduct approach to NSA see Wm. Hatcher: Calculus is Algebra, AMM 1982, and Van Osdol: Truth with Respect to an Ultrafilter or How to make Intuition Rigorous. For a much more comprehensive introduction to ultraproducts see Paul Eklof. Ultraproducts for Algebraists, 1977.
A: The heavy artillery of the model-theoretic approach to non-standard analysis is doing much more than just producing any one non-Archimedean ordered field or even a non-Archimedean ordered field which is elementarily equivalent to $\mathbb{R}$ (an ordered field is elementarily equivalent to $\mathbb{R}$ iff it is real-closed, so starting with any non-Archimedean ordered field, e.g. $\mathbb{R}(t)$, and taking the real-closure produces such a field).  There is an elaborate machinery of transfer, standard parts, etc. to overcome the fact that calculus does not work as we want it to in any non-Archimedean ordered field.
Note that the unique (up to unique isomorphism!) Dedekind-complete ordered field is $\mathbb{R}$, so any other ordered field is not Dedekind-complete.  More directly, if $K$ is a non-Archimedean ordered field, then the set of positive integers is bounded above (by any infinite element of $K$) but has no least upper bound, so $K$ is not Dedekind complete.  This spells doom for many of the basic results of real analysis.
For instance, for an ordered field $(K,\leq)$, the following are equivalent:
(i) $K$ is isomorphic to the real numbers.
(ii) $K$ satisfies the least upper bound axiom (i.e., $K$ is Dedekind complete).
(iii) Any bounded monotone sequence in $K$ is convergent.
(iv) (Bolzano-Weierstrass) Every bounded sequence in $K$ has a convergent subsequence.
(v) Any interval in $K$ is connected in the order topology.
(vi) Any closed bounded interval $[a,b]$ is compact in the order topology.
So working in a non-Archimedean field -- and not using model-theoretic cleverness to carefully restrict to certain kinds of statements -- calculus as we know it fails dramatically.
