Can be found in other languages.

Easier to read and shorter to write.

higher than the bitwise not (~) and unary minus,

The proposed associativity is right, just like how power towers work.

Removed turning pow() into a language construct from the proposal, due to BC breaks.

Should -3 ** 2 evaluate to 9 instead of -9 ?

According to the following resources, the scale tips more towards having the exponent precede the unary minus:

Similar languages

Ada

D

F#

Fortran

Freemat

Haskell

Mathematica / Matlab / Scilab

Octave

Perl

Python

R

Ruby

Sage

VB / Basic

Dissimilar languages

Bash

Cobol

ColdFusion

Excel

Tcl

Should 2 ** 3 ** 2 yield 64 (left associative), 512 (right associative) or throw an error (non associative)?

The exponent operator evaluation order should be based on Tetration and therefore be right associative.

Languages with left associative exponential operator

VB (not by choice imho)

Basic

Octave

Matlab

ColdFusion

Languages with right associative exponential operator

D

Haskell

R

F#

Ruby

Perl

Python

Mathematica

Freemat

Scilab

Tcl (changed from left associative!)

Cobol

Fortran

Sage

Bash

Languages with non associative exponential operator

Ada

Sources