`x = 32, y = 32, rule = 345/2-e/3`

3.A$2.A.B$.3A$A.A.A.A$.B.A.A.A2.B$4.A.5A.BA$3.A.A.A2.3A$4.3A.2A.A.A.B

$5.A.A.2A.A.A2.A$5.A.2A.2A.A.3A.B$4.B2A.2A.2A.2A.2A.A$6.2A.2A3.A.2A.A

.A$5.BA.A.A2.A4.A.A.B$5.A.A.A2.A.A2.A.A.A$8.A.2A.6A.4A$7.B.2A3.A2.A2.

A.A$9.A.A2.A2.A3.2A.B$8.4A.6A.2A.A$10.A.A.A2.A.A2.A.A.A$9.B.A.A4.A2.A

.A.AB$10.A.A.2A.A3.2A.2A$11.A.2A.2A.2A.2A.2AB$12.B.3A.A.2A.2A.A$14.A

2.A.A.2A.A.A$16.B.A.A.2A.3A$19.3A2.A.A.A$18.AB.5A.A$21.B2.A.A.A.B$25.

A.A.A.A$28.3A$27.B.A$28.A!

Chaotic growth starts from as little as two cells, a.la brian's brain:

`x = 1, y = 2, rule = 345/2-e/3`

A$A!

Speaking of that rule, there's the c/4 ship in this rule, although with an infinitely extensible backend:

`x = 8, y = 30, rule = 345/2-e/3`

.B$A$.BAB$2.A3$2.B$.B2.B$A2.2A$.BAB$2.A3$4.B$3.A$2.B2.AB$.B2.BA$A2.2A

$.BAB$2.A3$5.B$4.B2.B$3.A2.2A$2.B2.AB$.B2.BA$A2.2A$.BAB$2.A!

And this weird variant I found:

`x = 6, y = 6, rule = 345/2-e/3`

.A$BAB$3.A$4.B$4.2A$4.B!

There's also a c/2 diagonal, along with a sparkier interaction, and one possible puffer made from it:

`x = 20, y = 10, rule = 345/2-e/3`

18.BA$16.B.A$15.B.ABA$15.A.2A$15.B$4.B8.B$3.A.A6.A.A$2.A.B6.A.B$B.BA

5.B.BA$.3A6.3A!

But the main attraction is this lightspeed one-dimensional replicator:

`x = 11, y = 12, rule = 345/2-e/3`

.B$.A2.B3.A$.B4.2AB2A$B5.2AB2A$4.B3.A3$4.B3.A$B5.2AB2A$.B4.2AB2A$.A2.

B3.A$.B!

I'm surprised it even has an endpost, but here you go:

`x = 67, y = 16, rule = 345/2-e/3`

15.A$14.B.B$15.B$.A.A.A3.A.A3.A3.A.A3.A39.A$3A.ABA.ABA.B.ABA.B.ABA.AB

2A36.3A$.A2.ABA.ABA.B2.B2.B.ABA.AB2A37.A$5.A3.A2.B2.B2.B2.A3.A$12.B2.

B2.B$12.B2.B2.B$5.A3.A2.B2.B2.B2.A3.A$.A2.ABA.ABA.B2.B2.B.ABA.AB2A37.

A$3A.ABA.ABA.B.ABA.B.ABA.AB2A36.3A$.A.A.A3.A.A3.A3.A.A3.A39.A$15.B$

14.B.B$15.A!

It could probably be gunned, but save yourself the effort, there are two extremely common guns already:

`x = 8, y = 14, rule = 345/2-e/3`

.A$3A$.A$2.BA$2.BA$.A$3A$.A3$.A4.A$.6A$B.A2.A.B$.A4.A!

The latter is actually spawned usually from a wickstrecher, which can sometimes have this weird semi-signal on it:

`x = 11, y = 15, rule = 345/2-e/3`

8.BA$8.BA2$5.6A$5.A.AB2A$4.A3.A4$.AB5.BA$.AB5.BA2$11A$2ABA.A.AB2A$2.A

5.A!

Timebomb (starts exploding roundabout 875):

`x = 11, y = 7, rule = 345/2-e/3`

.AB5.BA$.AB5.BA2$11A$2ABA.A.AB2A$2.A5.A$4.A!

There's also a weird wickstrecher that can have chaos leech onto it:

`x = 23, y = 23, rule = 345/2-e/3`

4.B$3.A.A$2.A.B$B.BA$.3A16$21.BA$20.2BA$20.2A!

Longest lasting non-explosive methuselah I've found:

`x = 3, y = 3, rule = 345/2-e/3`

2A$A.A$.A!