Reading through the TeX Live manual, the best way to achieve what you want seems to be to alter where TEXMFHOME points to. This is where TeX looks for your personal files, and has standard setting ~/Library/texmf, as you probably know!
At the Terminal, typing
sudo tlmgr conf texmf TEXMFHOME "~/texmf"
will alter the location to ~/texmf, which would be the normal setting on Unix systems other than a Mac. Note that this alters the setting system-wide, but I'm guessing that for most people this will be acceptable.
If you want to have more than one possible location (say retaining the standard setting and adding another one), then a colon is used to separate the list:
sudo tlmgr conf texmf TEXMFHOME "~/Library/texmf:~/texmf"
By the way, for recent releases of TeX Live you don't need to run texhash for your local tree. The database is only required for the 'big' installation tree, while the local tree will be scanned when TeX runs. The assumption is that the local tree will always be relatively small, so this is an acceptable performance hit.
Apparently, this is (was) a bug in pgf 2.10 : I can reproduce it with pgf 2.10 and it works in the current pgf CVS.
As a workaround, I suggest to use sqrt(x) which seems to use a special implementation - it works in pgf 2.10 as well.
To really solve the problem, you may need to upgrade to a more recent pgf version (I believe there are still unstable pgf builds on texample.net (?) ).
If that is infeasible, you may want to explain more about your use-case such that others can comment on alternatives.
EDIT:
According to your comment, you really search for a more general solution without upgrading pgf. There is further solution. I did not mention it in the first place because I have never used it. The idea is to use the fpu library of PGF (maybe the fp library also does the job).
With fpu, the solution is as follows:
\documentclass{minimal}
\usepackage{tikz}
\usepgflibrary{fpu}%----- this
\begin{document}
\begin{tikzpicture}[scale=6,domain=0:1]
\draw[domain=0.0000125:1,
/pgf/fpu,/pgf/fpu/output format=fixed%------ this
] plot (\x,{(\x)^(0.8)-\x});
\end{tikzpicture}
\end{document}
I suppose this will always work. If replaces the internal PGF math engine (but only for that specific drawing command). The solution comes with minor changes to your documents and should not change the nature of your figures at all.
In case you are generally interested in plots (perhaps with more flexible plot handlers, 3d, or large data ranges, combined with a complete axis), you may also want to take a look at \usepackage{pgfplots}. This, however, changes your figures considerably.
Best Answer
If your TeXLive 2012 distribution for Mac OS X is MacTeX 2012, you should get a usable CVS version of TikZ/PGF with only two steps:
Downloaded the TDS package.
Extract the package into the root directory of a TDS tree used by your distribution. For example with MacTeX:
/Users/<username>/Library/texmf/./Users/<username>/Library/texmf/is the user's TDS tree and, by default, there is no need to callmktexlsr(texhashis an old name).To remove this CVS version, delete the following directories (where
.is your TDS directory):./source/latex/pgf./doc/generic/pgf./doc/pgf./scripts/pgf./tex/generic/pgf./tex/generic/pgf/graphdrawing/lua/pgf./tex/latex/pgf./tex/context/third/pgf./tex/plain/pgfNote: To get the latest complete version of TikZ, see my answer to How to install a current version of TikZ?