Just curious how to typeset letter 'a' in initex without 'overfull' warnings.

$ tex -ini '\font\f=cmr10\f a\end'
This is TeX, Version 3.14159265 (TeX Live 2015/Debian) (INITEX)

Overfull \hbox (5.00002pt too wide) detected at line 0
[]\f a
(see the transcript file for additional information)
Output written on texput.dvi (1 page, 180 bytes).
Transcript written on texput.log.

Just to show a slightly different approach, if you don't set the hsize you can avoid an overfull box by not starting a paragraph

tex -ini '\catcode`{1\catcode`}2\font\f=cmr10\f\hbox{a}\end'

There are a lot of basic values that are initialised by the plain format so are not set here. In particular, you need a non-zero \hsize and some \parfillskip stretch:

\hsize=10pt\parfillskip=0pt plus 1 fil\font\f=cmr10\f a\end

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