Updated: May 1, 2019
Of all the ways you can make a 13th chord, there are only two which you can voice with all the notes (i.e. root, 3rd, 5th, 7th, 9th, 11th and 13th, so they contain all the notes in the scale), without any nasty tritones or minor 9ths creating weird resonances.
Both are built from alternating minor 7th and Major 7th chords, and both have a nice, open, resonant sound.
The first is the brighter-sounding Major 13th, with #11, which is built on the Lydian mode. Here it is in G:
You can split the chord into Major 7th and minor 7th chords:
The darker, evil twin is the minor 13th chord, which is built on the Dorian mode.
Here it is in G:
It also splits into Major 7th and minor 7th chords, but with the minor 7th on the bottom:
The Whole Cycle
You can build this pattern into and endlessly rising chord series, moving around the cycle of fifths for all eternity.
As a chord progression it has the strange effect of constantly modulating, without seeming to ever move anywhere.