how many types of offspring might be produced in a Dihybrid cross?

The answer will vary depending on the type of inheritance (if complete dominance where a trait is either one ot the other characteristic or incomplete/codominance where an intermediate trait is formed) and whether you're talking about the genotypic or phenotypic ratio (which may be the same or different, depending on the type of inheritance).

The best way to figure this out is to make a Punnett square and count the results each way.  Say you do a cross between plants whith red (R) or white (r) flowers that can be tall (T) or short (t).  The parents would each be RrTt:

__|_RT_|_Rt_|_rT_|_rt_|
RT|
Rt_|
rT_|
rt_|

and the genotypic results would be:
RRTT (1)
RRTt (2)
RRtt (1)
RrTT (2)
RrTt (4)
Rrtt (2)
rrTt (2)
rrTT (1)
rrtt (1)

If the traits both showed complete dominance (the flowers would be red OR white and the plants tall OR short), you'd get the following phenotypes:

red tall (RRTT+RrTT+RRTt+RrTt) = 9
red short (RRtt+Rrtt) = 3
white tall (rrTT+rrTt) = 3
white short (rrtt) = 1

But if an intermediate "type" is produced by heterozygous individuals (Rr would be pink or red and white spotted or striped and Tt would be medium-height) you'd have the following numbers:

red tall (RRTT) = 1
red medium (RRTt) = 2
red short (RRtt) = 1
pink tall (RrTT) = 2
pink medium (RrTt) = 4
pink short (Rrtt) = 2
white tall (rrTT) = 1
white medium (rrTt) = 2
white short (rrtt) = 1

And what if only the color produced an intermediate trait and the height showed complete dominance?  

So, the answer will vary depending on what type of inheritance is shown by the specific traits being crossed.