I know one dot on the dotted half note means to play the note for 3 beats, but what do two dots mean:
12To belabor the clarification: This is not a case of a half note with two dots (which does happen); this is two half notes, each with one dot.– chrylis -cautiouslyoptimistic-Aug 3, 2020 at 4:39
It means that the two notes are dotted: both of them are 3 beats long.
This is the notation for chords: here are some examples here.
To briefly expand on @Tom_C's answer (thanks to @Guidot's comment below): There is no notation for lengthening all notes of a chord; each note has to be dotted individually.
Both dots are required, because there are situations where one plays two (or more) notes together but holds them for different lengths.
For example, here's an excerpt from Chopin's Prelude in A Minor, op. 28, no. 2 (m. 22)
On beat 3, the right hand plays D-E-G#-B. But whereas the lower three notes are each two beats long, the B lasts 1.5 beats and then moves to C.
There is such a thing as a double-dotted note. In this case, the dots will be placed side by side. The dot means "add half of whatever is to my immediate left. So, in this example from Chopin's Prelude in G Major, op. 28, no. 2 (m. 16), there is a quarter note (one beat) with a dot (half the quarter note) and then another dot (half of the preceding dot) for a total of 1.75 beats.
In other words,
X:0 T: Example: Double dots K:none M:none L:1/16 C7
is equivalent to
X:0 T: Example: Ties K:none L:1/16 C4-C2 -C1
Triple dots are possible as well. The same Prelude (G Major, op. 28, no. 3) in measure 9 has one.
In this situation
X:0 T:Example: Triple dots K:none L:1/16 C15
is equivalent to
X:0 T: Example: Tied equivalent K:none L:1/16 C8-C4-C2 -C1
The Chopin scores used in these examples can be found on IMSLP.
2Just adding on: when those situations arise, typically the notes are stemmed in opposite directions. Aug 2, 2020 at 19:14
You are typing too fast :P !– TomAug 2, 2020 at 19:23
@Tom_C: Oops! I've revised my answer to credit yours.– AaronAug 2, 2020 at 19:36
2I consider the first paragraph as overcomplicated explanation and suggest: since there is no notation for lengthening all notes of a chord, each note has to be dotted individually.– guidotAug 3, 2020 at 7:17
2There's even the very-rarely seen "half-dotted note", where the note is extended by 1/4 of its length. I've seen this represented as a dot with a diagonal slash through it. More often than not, though, they'll just write two notes tied together. Likewise for the "triple-dotted note", which are also somewhat rare. (My copy of that Chopin prelude has multiple tied notes rather than the triple-dot.) Aug 3, 2020 at 14:59
"What does it mean when a dotted half note has two dots instead of one?"
It means the note is lengthened by half its value, then by a further half of that. So a double-dotted half is a half plus a quarter plus an eighth.
But what you showed us isn't a half note with two dots. It's two half notes with a dot each.