I am not sure whether there are studies to the exact amount of aflatoxin that would lead to liver cancer. What I know are the levels which are allowed in foodstuff generally:
"In countries where a regulation for aflatoxin exists, tolerance levels for the total aflatoxin (sum of aflatoxins B1, B2, G1 and G2) ranges from 1 to 35 µg/kg for foods, with an average of 10 g/kg; and from zero to 50 µg/kg for animal feed, with an average of 20 µg/kg. For AFM1 in milk, tolerance levels are between 0.05 and 0.5 µg/kg, with most countries adopting a threshold of 0.05 µg/kg [10]."
Recent Trends in Microbiological Decontamination of Aflatoxins in Foodstuffs http://dx.doi.org/10.5772/51120
I doubt that such an exact figure is available as it would have required experiments on humans. Epidemiological studies always give a rough estimate as food intake data and toxin levels in the food, are always approximations.
The best carcinogenicity data come from animal experiments and then are extrapolated to humans. Safety factors are a big part in that extrapolation, so once again exact data will not be produced.
The theoretical answer is one molecule, as one mutation can be enough to produce a carcinoma.
If you are doing a risk assessment, the following publication from 2010, came up with a BMDL10 (bench mark dose associated with a 10% increase in liver cancer in Fisher rats, at the lower 10 percentile confidence interval) for aflatoxin B1 of 250 ng/kg bw/day. The work was part of a broader effort to illustrate the application of the margin of exposure (MOE) risk assessment approach to genotoxic/carcinogenic substances in foods. For aflatoxin B1, they found that human dietary exposures were about 100 to 600 fold lower than the BMDL10 (MoE range of 100 to 600). Assuming a simple linear mathematical relationship, that would work out to a theoretical cancer risk level of about 1-in-1000 to 1-in-6000. So that would be 1 theoretical additional liver cancer from the diet attributable to aflatoxin in 1000 to 6000 exposed "individuals". However, as pointed out by the previous responder, extrapolation from animal data to humans can introduce a lot of uncertainty.
Citations are below.
Benford et al., Application of the margin of exposure (MoE) approach to substances in food that are genotoxic and carcinogenic: example: aflatoxin B1 (AFB1). Food Chem Toxicol. 2010 Jan;48 Suppl 1:S34-41. doi: 10.1016/j.fct.2009.10.037. Review. PubMed PMID: 20113853.
Benford et al., Application of the Margin of Exposure (MOE) approach to substances in food that are genotoxic and carcinogenic. Food Chem Toxicol. 2010 Jan;48 Suppl 1:S2-24. doi: 10.1016/j.fct.2009.11.003. Review. PubMed PMID: 20113851.