Ikleyn gave us one possibility above, 144.  Let's see if we can find any
others.

So he's at least 100 years old!

Let h = the hundreds digit
Let t = the tens digit
Let u = the units or ones digit

His age = 100h+10t+u
His age = htu(h+t+u)

      100h+10t+u = htu(h+t+u)
(99+1)h+(9+1)t+u = htu(htu) 
    99h+h+9t+t+u = htu(h+t+u)
    99h+9t+h+t+u = htu(h+t+u)
          99h+9t = htu(h+t+u)-h-t-u
        9(11h+t) = htu(h+t+u)-(h+t+u)
        9(11h+t) = (h+t+u)(htu-1)

Pretty complicated.  Let's see if we can find any possibilities where he is
less than 200 years old.  That would mean that h=1. Substituting:

        9(11+t) = (1+t+u)(tu-1)

Let's see if there are any possibilities where the 1st and 2nd factors on
the left are equal respectively to the 1st and 2nd factors on the right.
[That is not necessarily the case, but it could be, and trying it may lead
to more answers].

   9 = 1+t+u, 
11+t = tu-1

Solve the first equation for u
 8-t = u

Substitute in the second equation

      11+t = t(8-t)-1
      11+t = 8t-t2-1
  t2-7t+12 = 0    
(t-4)(t-3) = 0

t-4=0; t-3=0
  t=4;   t=3

If t=4, then 8-4 = u = 4.  That gives us 144, which Ikleyn found above.
We don't know how she found it.  She just gave it.

But we also have another possibility for t=3

If t=3, then 8-3 = u = 5.  That gives us the age of 135, which Ikleyn did
not give above. 

The sum of the digits is 1+3+5 = 9 and the product is 1∙3∙5 = 15
and 9∙15 = 135.

So now we know that two possibilities are that the man is 135 or 144.

I conjecture that 135 and 144 are the only possible solutions.

In either case, he is older than any person has ever known to have lived to
be. Jeanne Louise Calment (1875-1997) from France lived to be 122+. She is
listed in Guinness's world records as having the longest life span on
record. 

Edwin