In a comment to my recent question on the origin of the two forms of sigma, Rafael pointed out that in Arabic most letters have separate forms for initial, middle, and final positions. However, in Greek only sigma seems to have different forms for different positions. Did any other Greek letters ever have different forms for the end of a word? In the Greek I know sigma is unique in this respect, but was this always the case?
The following is from the 'Variant forms' section of From Unicode to Typography, a Case Study: the Greek Script, by Yannis Haralambous (p 7–9). The paper was presented at the 14th International Unicode Conference (Boston, 1999). In the following instances, different letterforms have, at some times and in some places, been used, depending on the letter's position in the word. (I've added some bold font for presentation purposes and italic for emphasis.)
The beta without descender: ϐ. Sometimes the following rule is applied: β is used at the beginning of words, and ϐ otherwise, as in the words βλάϐη, βιϐλίο, βάρϐαρος. This rule seems to be geographically very restricted: the author has seen it applied in Greek books of the late 19th and early 20th century...; it seems to be applied, rigorously and continuously until today, only in France... all Budé and Le Cerf editions of classical Greek texts apply it as well. On the other hand, this rule is unknown in the rest of the world....
The "open" and "closed" letter pairs theta and phi: ϑ/θ, φ/ϕ (Unicode U+03D1, U+03B8, U+03C6, U+03D5). These variant forms are mentionned in most grammars and dictionaries, but the author was not able to detect any contextual rule involving them. It is interesting to note that [Imp, p. 95] suggests that the open theta ϑ be used at the beginning of words; this rule is mentioned only by [Imp] and the author has never seen it applied.
The "round" pi ϖ (U+03D6). This letter has been a variant of π from the very first days of Greek typography: in Figure 2, the reader can see an excerpt of...a book printed in 1489 (!). In this excerpt both versions of pi have been used, apparently randomly.
Once again, [Imp, p. 95] suggests that the round pi be used only at the beginning of word. According to [Tre], this rule has been applied in certain French journals (for example the Mémoires de la Société de Linguistique in the late 19th and early 20th century) ; [Tre] applies it in all of his writings...
Here are Haralambous's bibliographical references for the preceding:
[Imp] = Lexique des règles typographiques en usage à l'Imprimerie nationale, Imprimerie nationale, 1990. (Typographical guide.)
[Tre] = Tremblay, Felix, personal communication
In addition, Haralambous discusses the variant forms of uppercase upsilon (ϒ/Υ), lowercase kappa (κ/ϰ), and lowercase rho (ρ/ϱ). In these instances, though, the variations depend merely on the typeface or font, and seem never to depend on position in the word. He also discusses the so-called lunate sigma (Ϲ, ϲ); as far as I'm aware, if lunate sigma is used at all, it's used for all sigmas in all positions, both uppercase and lowercase.
So, although only sigma has a different form for the end of a word vs. the beginning or middle (unless lunate sigma is used for all sigmas), in the typographical practices of certain places and/or certain times, some letters have different forms for the beginning of a word vs. the middle and end.
Beta comes to mind. From wikipedia
In some high-quality typesetting, especially in the French tradition, a typographic variant of the lowercase letter without a descender is used within a word for ancient Greek: βίβλος is printed βίϐλος,
As I hope you can see in the above quote, there is even a special Unicode point for it, incorrectly named GREEK BETA SYMBOL.
How is this case different from the many variant glyphs listed in @cnread’s answer from Haralambous’ excerpt (ϖ, φ, ϕ, ...)? While in some scientific notations those variants are considered different symbols, and, as documented by Haralambous, there were attempts to establish typographical rules about their usage, in fact such attempts mostly failed, and, when it comes to writing Greek texts as opposed to mathematical formulas, they remained a matter of font design, like roman vs. italic (even if it is funny to use those names for Greek letters...)
On the other side, in the case of β vs. ϐ a rule was established and applied consistently for decades, even if only in French editions of Greek Classics and, for a short period, in Greece itself.
In fact, in early Greek typography, a huge number of ligatures and variant forms were taken over from handwiting. In the course of time, most of those ligatures disappeared.
It is enough to visit some Greek websites to see that nowadays, in Greece itself, everybody uses β, Unicode U+03B2 (or some equivalent 8-bit encoding) and leaves it to the font to decide if there has to be a descender or not. Normal Greek keyboards don’t have a simple way to produce ϐ, U+03D0, so perhaps it is not so incorrect to name it SYMBOL, pace those French typesetters.
added for completeness
Epsilon also has two lowercase forms: ε and ϵ (also called lunate, just like the rounded sigma, or uncial.) The distinction seems to be mainly one of periods, not position. Uncial form was apparently predominant between IV and VIII centuries AD.
Interestingly enough, in medieval times, many letters developed not end-of-word variants, but a number of ligatures.
This book (p. 191 ss.) shows different forms of Greek letters by centuries along the I millenium AD