How many human cells are there in our body, on average?
Bianconi et al. 2013 give an estimated lower bound of 3.72 × 10^13 (which, by the way, is approximately the geometric mean of 10^13 and 10^14).

However, from the table in their Supplemental Information (where estimates for about fifty different types of cells are added up), it is clear that the vast majority of these are the erythrocytes, also known as red blood cells: estimated at 2.63 × 10^13. While the second largest class are the glial cells (from the nervous system), estimated at 0.30 × 10^13. Three other classes (endothelial cells {vessels}, dermal fibroblasts {skin}, and platelets {blood}) add up to another 0.58 × 10^13. These, in total, is about 3.5 × 10^13, and we can stop there, because the uncertainty in all these estimates swamps all the other cell type populations, which are much smaller.

This means that if we only consider the 5 most abundant cell types in the body, we already get a very good estimate. Is it likely that we have missed a cell type of which there are a trillion cells in the body? It's not impossible, but unlikely, because it would be hard to miss all these cells.

Maybe there is a long tail (i.e. a large number of minor cell types, each with a small, but significant population) that wasn't considered? Together, these could add up to a significant adjustment to the above figure. I don't know.

This estimate is for a typical human person: 30 year old young adult, 70 kg, 1.72 m tall and with a surface area of 1.85 m^2. For this person, the figure is given with roughly a +-20% uncertainty.

Given all the above, it's probably safe to say "30 to 50 trillion".

(This obviously doesn't take into account any of the bacteria mentioned above.)