RequirePackage{filecontents}
begin{filecontents*}{jobname.bib}
    @article{a_paper,
        author = "N. E. Body and A. N. Other",
        title = "An article about {Al}$_x${Ga}$_{1-x}${N}, {GaN}, {In}{$_x$}{Ga}{$_{1-x}$}{N} and other compounds with Annoying Names {$omega$}",
        journal = "Journal of tricky examples"
    }
end{filecontents*}
documentclass{article}
usepackage{biblatex}
addbibresource{jobname.bib}

%sentence case for biblatex
DeclareFieldFormat{sentencecase}{MakeSentenceCase{#1}}
renewbibmacro*{title}{%
  ifthenelse{iffieldundef{title}ANDiffieldundef{subtitle}}
    {}
    {ifthenelse{ifentrytype{article}ORifentrytype{inbook}%
      ORifentrytype{incollection}ORifentrytype{inproceedings}%
      ORifentrytype{inreference}}
      {printtext[title]{%
        printfield[sentencecase]{title}%
        setunit{subtitlepunct}%
        printfield[sentencecase]{subtitle}}}%
      {printtext[title]{%
        printfield[titlecase]{title}%
        setunit{subtitlepunct}%
        printfield[titlecase]{subtitle}}}%
     newunit}%
  printfield{titleaddon}}

begin{document}
    Test the complete texttt{.bib} file: nocite{*}.
    printbibliography
end{document}

! Missing { inserted.
<to be read again>
                   }
l.35 end
         {document}
?
! Missing } inserted.
<inserted text>
                }
l.35 end
         {document}
?

(./chrisbib.aux) (./chrisbib.bbl)
! Missing { inserted.
<to be read again>
                   }
blx@tempa ...ercase {$_x$}{Ga}MakeLowercase {$_}
                                                  {1-x}MakeLowercase {$}{N}...
<argument> ...{csname abx@field@titleendcsname }
                                                  blx@endunit
@secondoftwo #1#2->#2

<argument> printfield [sentencecase]{title}
                                            setunit {subtitlepunct }print...

@article{a_paper,
  author = "N. E. Body and A. N. Other",
  title = "An article about {Al$_x$Ga$_{1-x}$N}, {GaN}, {In$_x$Ga$_{1-x}$N}
           and other compounds with Annoying Names {$omega$}",
  journal = "Journal of tricky examples"
}