I think, though I’m not certain, that this letter to the journal Nature is accessible without a subscription.  The author, Mikhail Gelfand, is responding to a remark in an earlier issue that Russian had the largest alphabet of any language, and points out that several of the languages of the Caucasus have still larger ones.

“The outright winner is the Archi alphabet, developed in 2006. This is another language from the Caucasus and has 97 letters — although many of these are groups of two, three or even four or five characters, rather than independent signs. The highest number of independent signs, at 41, is probably to be found in Abkhaz.”

Inaccessible without a subscription, alas. It sounds interesting. I wonder if some of these large alphabets count diphthongs as characters. (Ah, I guess the ‘four or five characters’ answers that. The English alphabet could considerably enlarge itself if it did likewise.)

Well, it’s only 3 paragraphs long.  I’m going to paste the whole thing here, but I promise not to sulk if Dave deletes it for copyright reasons.

Nature 454, 691 (7 August 2008) | doi:10.1038/454691a; Published online 6 August 2008

Some alphabets easily beat Russian letter count

Mikhail S. Gelfand1

1. Institute for Information Transmission Problems, B. Karetny pereulok 19, Moscow 127994, Russia
Email:

Sir

In his Essay about the Phaistos Disc (’A century of puzzling’ Nature 453, 990–991; 2008), Andrew Robinson notes that the largest known alphabet is Russian, with 36 letters. In fact, the Russian alphabet has had 33 letters since 1918; before that, it officially had 35 (37 in reality).

But even this number falls short by comparison with the alphabets of the many consonant-rich languages of the northern Caucasus. These commonly have more than 40 letters (for example, there are 45 in Lezghin, 49 in Chechen, 51 in Avar and 62 in one of the dialects of Abkhaz).

The outright winner is the Archi alphabet, developed in 2006. This is another language from the Caucasus and has 97 letters — although many of these are groups of two, three or even four or five characters, rather than independent signs. The highest number of independent signs, at 41, is probably to be found in Abkhaz.

Archi has 24 affricates, surely a world record as well, and (according to Mikael Parkvall’s very enjoyable Limits of Language) is “a good candidate for the ‘most unexpected suppletive item’ category. The word bič’ni translates as ‘corner of a sack’, provided we are talking about one corner only. Several corners of sacks, however, is boždo.”

Would the ‘groups of characters’ be something like the CH and LL digraphs that Spanish had until recently? That would give an alphabet with 29 members (A-Z, Ñ, CH and LL) but only 26 or 27 distinct characters, depending whether Ñ is counted as distinct or as an accented N.

Nature 454, 691 (7 August 2008) | doi:10.1038/454691a; Published online 6 August 2008
...
The outright winner is the Archi alphabet, developed in 2006.

Various Internet sites show that there are 1000 to 1200 speakers of this language.  I can believe that linguists might invent a new alphabet to help record the language, but do you suppose the people have actually widely adopted it? 
The residents in the old age home were probably thrilled to find out they had to learn 60 new letters in order to read the cafeteria menu from now on.  (That’s a joke.)

So which language has the greatest number of actual phonemes?

Dr. Techie - 03 October 2008 07:57 AM

The highest number of independent signs, at 41, is probably to be found in Abkhaz.

I don’t understand this. Devanagari, used to write Sanskrit, has more. I suppose one might argue that the long versions of vowels aren’t independent characters, but length is indicated in different ways; it isn’t analogous to a single symbol like the bar used in some English phonetic transcriptions above any vowel to show that it’s long.

http://www.omniglot.com/writing/sanskrit.htm

It seems to me that the top row on that page, for example, which shows 17 vowels, really has only 6 or at most 7 “independent” symbols.

It depends on what is called an “alphabet”.

This is a bag of worms.

Here is a Wiki piece ...

http://en.wikipedia.org/wiki/Writing_system

... which distinguishes (in addition to “true alphabets") the so-called abjads and abugidas ... sometimes called “consonantal alphabets”.

There are of course also syllabaries ... and the Korean system which can (IMHO) be regarded as exemplifying either an alphabet or a syllabary.