Adding symbols to the beginning of certain references in the reference list

You might do this using the biblatex 3.4+/biber 2.5+ Data Annotation feature although the linked article from @Guido's comment is also a good solution since you are annotating entries rather than fields in an entry.

\documentclass[man]{apa6}
\usepackage{filecontents,showframe}
\usepackage[american]{babel}
\usepackage{csquotes} 
\usepackage[style=apa]{biblatex}
\DeclareLanguageMapping{american}{american-apa}
\usepackage[T1]{fontenc}
\shorttitle{This is the title of the article}

\begin{filecontents*}{mybib.bib}
@article{ID1,
    Author = {Some Author},
    Author+an = {=ref},
    Journal = {Journal of Non-existence},
    Title = {Being on the top: How it feels},
    Year = {2014},
    Volume = {1},
    Pages = {1-10},
    Keywords = {Reference}}

@article{ID2,
    Author = {Some Dufus},
    Author+an = {=db},
    Journal = {Scandinavian Journal of Bogus},
    Title = {Second to none (except to one)},
    Year = {2015},
    Volume = {2},
    Pages = {11-20},
    Keywords = {Database}}

@article{ID3,
    Author = {Some Writer},
    Author+an = {=data},
    Journal = {Journal of Fake},
    Title = {Number number three: How it feels},
    Year = {2016},
    Volume = {3},
    Pages = {21-30},
    Keywords = {Data received via e-mail}}

\end{filecontents*}
\addbibresource{mybib.bib}
\renewbibmacro*{name:hook}[1]{%
  \iffieldannotation{db}
    {*}
    {}%
  \iffieldannotation{data}
    {**}
    {}}
\begin{document}
\nocite{*}
\printbibliography
\end{document}

You can define a new entry option extsymb where you can give an arbitrary symbol to be printed before the entry in the bibliography

\makeatletter
\providecommand{\bib@extsym}{}
\DeclareEntryOption[string]{extsym}{\renewcommand{\bib@extsym}{#1}}
\renewbibmacro*{begentry}{\printtext{\bib@extsym}}
\makeatother

this is used as

options       = {extsym={*}},

in the .bib entry.

MWE

\documentclass[american]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[backend=biber, style=apa]{biblatex}
\DeclareLanguageMapping{american}{american-apa}

\usepackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@book{bohec,
  author        = {Le Bohec, Yann},
  title         = {Histoire militaire des guerres puniques},
  date          = {1996},
  location      = {Monaco},
  publisher     = {Rocher},
  isbn          = {2-268-02147-5},
  options       = {extsym={**}},
}
@book{uthor,
  author        = {Uthor, Arnold},
  title         = {A Book},
  date          = {2013},
  location      = {Place},
  publisher     = {P. Ublisher's \& Co.},
  options       = {extsym={*}},
}
\end{filecontents*}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}

\makeatletter
\providecommand{\bib@extsym}{}
\DeclareEntryOption[string]{extsym}{\renewcommand{\bib@extsym}{#1}}
\renewbibmacro*{begentry}{\printtext{\bib@extsym}}
\makeatother

\begin{document}
  \cite{wilde,cicero,coleridge,vangennep,bohec,uthor}
  \printbibliography
\end{document}

---

You can of course also extend the methods from Functionality of apacites \nocitemeta with biblatex-apa: adding asterisks to author lastnames (meta-analysis)

\DeclareBibliographyCategory{asterisk}
\DeclareBibliographyCategory{doubleasterisk}
\renewbibmacro*{begentry}{%
  \ifcategory{asterisk}%
    {*}%
    {}%
  \ifcategory{doubleasterisk}%
    {**}%
    {}%
}

Where you then have

\addtocategory{asterisk}{uthor}
\addtocategory{doubleasterisk}{bohec}

in your .tex file.

---

Lastly, you can use keywords

\renewbibmacro*{begentry}{%
  \ifcategory{asterisk}%
    {*}%
    {}%
  \ifcategory{doubleasterisk}%
    {**}%
    {}%
}

with

@book{uthor,
  author        = {Uthor, Arnold},
  title         = {A Book},
  date          = {2013},
  location      = {Place},
  publisher     = {P. Ublisher's \& Co.},
  keywords      = {asterisk},
}

in the .bib file.