We are interested text mode, not math. First, with the default encoding OT1:
\l \aa \$ \textasciicircum cannot take accents on top. There are probably others. However, they can take accents below, e.g., cedilla and underbar.
\ae \o can take accents on top.
Accents above \textbullet are off center. Those below look fine.
Cedilla on characters with descenders, j, upside down exclamation, all the braces and parentheses, look terrible.
Ogonek is not available unless \usepackage[T1]{fontenc}.
The Great, Big List of LaTEX Symbols David Carlisle, Scott Pakin, Alexander Holt, February 7, 2001. (It is missing the undotted i and j.)
Text mode greek requires additional packages.
After \usepackage[T1]{fontenc} , \l, \aa, \$, \textasciicircum can take accents on top. < no longer encodes upside down exclamation mark. One can put two circles on top of the a.
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