This one, Gijswijt's sequence, I thought was particularly interesting rhythmically, so suitable to share here on the Bounce Metronome blog.

As you see it plays 4s occasionally, and you wonder if it will ever play a 5. Well, yes it will apparently, but only after it has played 10^(10^23) notes. So in practise, won't ever reach that in the foreseeable future no matter how long it is played for, or how fast the tempo, if played from the beginning all the way through.

ABOUT THE TUNING

It's a major chord. You may wonder if you hear the 4 played faintly sometimes between the other notes, but no it isn't. That's just because it is a harmonic of the lowest notes. The pitches played are 3/2, 2/1, 5/2, 6/2, and if a 5 was ever played it would be a  7/2. The faint "4"s you might think you hear is the second harmonic of the 3/2 sounding like a 6/2.

HOW THE SEQUENCE IS DEFINED

The nth number in the sequence gives the number of repeating blocks in the sequence so far, where you are allowed to ignore the beginning of the sequence, as much of it as you like, but it must then repeat exactly from then on.

So for instance, in:
1, 1, 2, 1, 1, 2, 2, 2, 3,
the first few 2s and the 3 are found like this
(1 ; 1 ) 2
1 1 2 (1 ; 1) 2
(1, 1, 2 : 1, 1, 2), 2 
1, 1, 2, 1, 1, (2 ; 2) 2
1, 1, 2, 1, 1,( 2 ; 2 ; 2), 3,
where the repeating blocks are shown in brackets separated by colons.

This is the first "interesting" 3:
1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, (2, 2, 2, 3 : 2, 2, 2, 3 : 2, 2, 2, 3), 3,

To find out more, and to listen to Van Eck's sequence as well, and some Sloth Canon sequences from Tune Smithy, visit: Tune Smithy Melodies for other sequences in the OEIS.

More about 10^(10^23) (the number of notes you have to play in this sequence before you get a 5)

It's a number consisting of 1 followed by 100,000,000,000,000,000,000,000 zeroes.

You could write this number out  even on the surface of the Earth as a decimal expansion though the individual digits would have to be much smaller than millimeter sized. I make it that each zero would be about 40 microns across if you inscribed this number on the available land surface of the Earth (ignoring oceans) :). In other words, the digits would be about a 25th of a millimeter in height, a little taller as they are narrower than they are high but smaller than a 10th of a millimeter surely in an easily readable font.

So, it's not like a googolplex, too large to even write out within the observable universe in decimal even using one atom for each digit - but it is pretty vast :).

For mathematicians, though it is vast, yet this isn't regarded as a remarkably enormous number nowadays. Far far larger numbers have been used in maths, some too large to write out within the observable universe even using exponentiation, such as Grahams number.