Recently I've come across some double-dotted notes, and even some triple-dotted ones which seem a bit quirky to me, but who knows?
For one dot, you add 1/2 of the note's length, so I suppose for two dots, you add 1/2 + 1/4, and for three dots, you add 1/2 + 1/4 + 1/8.
Is the practice of using more than one dot common in modern scores? It seems to me they appear more in older works. Also, this seems a bit silly, but could you potentially dot a note with more than three dots?
Dots in general start to get messy after the first one and can lead to confusion to while sight reading if more than one is used. For the sake of sight reading there are even some syncopated lines where a normal duration like a quarter note or eighth note are represented as ties to show the beat better.
Using more than one dot is more theoretical in nature than practical. I know I've only used a double dot once in actual peice, but eventually changed it because it was easier to follow.
You can have as many dots as you like, but it gets crazy pretty fast. Here's a quintuple dotted half note and a 128th note to fill out the last part of the measure and the equivalent tied:
You'll probably never encounter anything close to this, but I'm sure you would prefer to read the tied version instead as it is a lot clearer.
I'm guessing that tied notes have rather taken over. They're easier to read - were there two or three dots?- and the grouping probably is easier to follow. Let's face it, it's simpler to read a crotchet tied to a shorter note than do the sums to work out how long the (double) dotted note needs to be.
Historically, up to the 18th century a single dot followed by short notes didn't necessarily mean "add exactly half the length", It could mean either more or less than half, depending on the style and tempo of the music (and the performers were expected to be able to judge that without any written instructions on the score). You often see "mathematically incorrect" notations like a dotted quarter note followed by three 32nd notes.
In the early classical period (up to early Beethoven) dotted notes were sometimes used instead of ties across barlines - for example a half-note at the end of a bar and the dot written at the start of the next bar, where the tied note would be in modern notation.
In the 19th century the usage corresponded closer with the mathematical "add half" rule. Double dots were often used, and three dots occasionally. I can't remember ever seeing more than three.
Modern scores, especially with time signatures that are not so "simple" as 2/4, 3/4, 4/4, or 6/8 (or no time signature at all), tend to use tied notes rather than multiple dots. Counting "beats" in a time signature like 11/16 is hard enough without having to read multiply-dotted notes.
Note also that dotted rests have never been used, except where they represent whole beats in compound times - i.e. dotted quarter-note rests in 6/8 and 9/8, and dotted half-note rests for half a bar of 12/8 - though in the 18th century the dots after the such rests were often omitted, since the absence of any notes was enough to make a common-sense interpretation of what the notation meant.
Of course new editions of old music often change the original notation to match current notation conventions - which can sometimes add a misleading sense of "mathematical accuracy" which the composer never intended.
Instead of thinking of dots as being additive, think of dotted notes as being short of the next longest duration.
