First off, let us try to clarify a few terms:
- As usual in astrophysics, metal-free star means atomic number $Z \leq 3$, i.e. it only consists of the primordial elements hydrogen, helium, and lithium.
- Primordial star literally means original star and refers to the first star(s) (generation) formed after the big bang. It is IMHO equivalent to metal-free, and a primordial metal-free star would be a pleonasm.
- Star can be most generally be defined as massive, self-luminious gas sphere (see Lexikon der Astronomie, p. 412), which leads to the question what each of these four terms mean exactly. Massive would probably mean at least about $0.09 M_\odot$ as 0.09 solar masses is the weight of the smallest star observed, AB Doradus, which is undergoing nuclear fusion.
- A quasi-star is also worth a definition for the current question. The first paragraph of that Wiki-page summarizes it pretty neatly:
A quasi-star (also called black hole star) is a hypothetical type of extremely massive and luminous star that may have existed early in the history of the Universe. Unlike modern stars, which are powered by nuclear fusion in their cores, a quasi-star's energy would come from material falling into a black hole at its core.
ProfRob already pointed out that a key issue is the definition of star.
Or is something that starts nuclear fusion, but is always collapsing, a star?
Related is the question about the stability of a star, which is studied by Isabelle Baraffe et al. in arXiv:astro-ph/0009410 :
The stability of metal-free very massive stars ($ Z= 0; M = 120 \ldots 500 M_\odot$) is analyzed and compared with metal-enriched stars. Such zero-metal stars are unstable to nuclear-powered radial pulsations on the main sequence, but the growth time scale for these instabilities is much longer than for their metal-rich counterparts.
The metal-free stars analyzed in that manuscript would still be smaller than the quasi-stars, which require at least $1000 M_\odot$. Again a Wikipedia-quote:
Quasi stars would have had a short maximum lifespan, approximately 7 million years, during which the core black hole would have grown to about $10^3 \ldots 10^5 M_\odot$ for modern stars.
To summarize: The maximal size limit varies, depending on what we exactly regard as primordial metal-free star, and also how long such an object has to exist/ be stable. I would assume that the upper limit of primordial metal-free stars is probably larger than the upper limit of stability of $80 \ldots 100 M_\odot$ which holds for modern stars (see p. 458 in Lexikon der Astronomie).
- Helmut Zimmermann, Alfred Weigert: Lexikon der Astronomie. Edition 8. Heidelberg/ Berlin 1999. ISBN 3-8274-0575-0. (in German)