There was recent question on TeX.SX about expansion, where I suggested using \romannumeral for full expansion. This idea comes up quite often, so I thought it would be useful to look at how it works.
The \romannumeral primitive was intended to turn integers into roman numerals. As such, the input to \romannumeral should be an integer, in the same way that the input to \number is an integer. However, while \number produces output for both positive and negative integer input, \romannumeral produces no output at all for valid negative integer input. That is pretty clear with a simple case such as
\romannumeral -15
but becomes more powerful when used along with the ` syntax which TeX allows for including an integer
\romannumeral -`0
Here, the `0 is converted into a integer which TeX treats as complete: `0 is 48, but `01 is the (terminated) integer 48 followed by a separate 1 and not the integer 481. The important thing for expansion, however, is that TeX always looks for an optional space to gobble after an integer, even in a case like `0 where the integer is automatically terminated.
How does this help with expansion? It’s all to do with how TeX terminate numbers. If we have the demonstration macro
\def\demo#1{%
\detokenize\expandafter{\romannumeral-`0#1}%
}
the \romannumeral will produce no output (as the number it finds is −48). However, it will expand #1 until it finds an unexpandable token. That means that something like
\def\testa{\testb}
\def\testb{\testc}
\def\testc{abc}
\demo{\testa}
will be expanded to abc without needing to know in advance how many expansions are needed. In a real case, this is a great way to fully-expand \if... tests and so on, leaving only the result, but without needing to know how many expansions are needed (and having long \expandafter runs).
What you should notice here is that TeX will reinsert the expanded result, and will not complain if something non-expandable appears
\def\testa{\relax}
\demo{\testa}
will print \relax, as this is inserted with no errors or loss of unexpandable tokens.
The only proviso here is that TeX stops at the first non-expandable item. So something like
\def\testa{ \testb}
\def\testb{123}
\demo{\testa}
will stop at the space in \testa and not expand \testb. For real applications, that is not usually an issue as we are usually aiming to expand the beginning of the input.