How many people apply for a math postdoc in the US?
Sounds like a Fermi problem!

Here's my estimate.

How many new math PHDs are granted in the US per year?


Number of major US institutions (~1000)
Fraction of major US institutions offering a PHD math program (1 in 4, or 0.25)
Average number of doctorates conferred by each PHD math program per year (6)


This gives roughly (1000 x 0.25 x 6) = 1500 new math PHDs per year. 

How many postdoc positions are in the US?


Number of US institutions offering postdoc positions (300?)
Number of postdoc positions per offering institution (3?)
Fraction of postdoc positions opening per year (1 in 2, or 0.5) (assuming two-year positions)


That gives (300 x 3 x 0.5) ~ 450 postdoc positions opening per year

How many people apply to a given position?


Number of new math PHDs per year (1500, see above)
Fraction of new PHDs applying for postdoc positions (1 in 2, or 0.5)
Average number of postdoc applications each new PHD fills out (30?)
Number of open postdoc positions per year (450, or almost 500, see above)


This gives (1500 x 0.5 x 30 / 500) ~ 60 applications per postdoc position. 

This approach misses something, however.

Not everyone applying for these positions has just received their PHD: competition is increasing. There are fewer tenure-track positions open, causing postdocs to apply to a second (or even third!) postdoc position. Furthermore, not all applicants earned their PHD in the US. 

To account for all the other applicants (current postdocs, PHDs from other disciplines, etc.), let's multiply that figure by a factor of 1.5, so there's (60 x 1.5) = 90 applicants per postdoc position. 

Thus my final estimate:

I estimate that each postdoc position receives an average of 90 applicants per postdoc position. 

Accuracy

This PDF has some great figures (2013-2014) for comparison, though I checked this after my estimate:  


1926 math PHDs conferred. 
38% (626) are reported to be in postdoc positions in the US (another 200 or so are abroad)


Lastly, the "800 applicants for 2 positions" figure (comment from @Sana) is for an R1 institution, and is therefore likely to be much higher than it would be for an average institution.