Why should the first person pronoun 'I' always be capitalized?

The pronoun I began to be 'capitalized' around the middle of the 13th century. But this was not true capitalization. Note that it was long before the printing press: all texts were in manuscript.

Before the 11th century, the letter i was normally just a short vertical line, without a dot, somewhat like ı. The j did not exist as a separate letter. When an ı was written as a separate word or mark, as the Roman numeral ı/I and the pronoun ı/I, or when it was the last one of a group of ı's, it began to be written elongated, somewhat like a straighter ȷ (without a dot). This elongation of the separate, single ı was probably done in order to avoid confusion with punctuation marks. That of the last ı of a group was mostly in order to avoid confusion between u and ıı, between n and ıı, and between m and ııı, which often look identical in manuscripts; both m and ııı could be written with and without clearly distinguishable connecting strokes. From then on, such groups of ı's looked more like ıȷ and ııȷ (without dots).

I believe that this convention of elongating the pronoun I had already been established by the time the dot was first used. Because a long ȷ without a dot looks much like a capital I—which has been written the same way since Antiquity—, it was later assumed to be a capital. (Incidentally, the dot was then usually written as a very short diagonal line above the ı or ȷ.)

From Etymonline:

The reason for writing I is ... the orthographic habit in the middle ages of using a 'long i' (that is, j or I) whenever the letter was isolated or formed the last letter of a group; the numeral 'one' was written j or I (and three iij, etc.), just as much as the pronoun. [Otto Jespersen, "Growth and Structure of the English Language," p.233]

An illustration of the problem of indistinguishable ııı, uı, m, etc.:

Cedet animam meam in
te mee: dimittam adver
sum me eloquium meum loq[ua]r
in amaritudine anime mee di

[From Mechanical Snail's comment below:] By contrast, "i" (meaning "and") is not capitalized in Catalan / archaic Spanish, nor in Polish/Serbo-Croatian.

[From Janus's comment below:] Possibly related is the fact that the pronoun I in Danish (where it means ‘ye’, i.e., non-formal second person plural) is also always capitalised. The homophone i (which means ‘in’), however, is not.