Is there a reliable estimate of the number of cells on Earth?
Okay so I looked around a bit, and there are surprisingly few papers related to this question.

This paper is the best I cold find, but it's almost 20 years old. There's also this one, which slightly updates the other one, but they both focus on bacteria. For those the total number is estimated to be somewhere in the range of $10^{30}$.
For humans the latest published number is $3.72 \times 10^{13}$, so even with a few billion in total ($10^9 \times 10^{13} = 10^{22}$) we probably don't even matter for the total cell count. Since smaller more abundant animals have a much lower number of cells, they probably don't matter either.

Now to the only thing I'm not sure about: plants

The number of plants on earth is really elusive, only more elusive is the number of cells per plant. This study claims around 3 trillion ($10^{12}$) tree's on earth. Since tree's are bound to have (many) more cells than humans, we're probably getting closer to the number of prokaryotic cells by now (I guess the number of 'tree cells' will be somewhere around $10^{27}  - 10^{30})$.
So far I didn't find anything about grasses or other plants (neither numbers nor cell counts), so it's really hard to say whether these matter or not.
This study seems to give an average number of cells per growth ring in tree's, but I'm not sure how to get a number of cells / tree from that.

In total the estimated number of bacteria ($10^{30}$) is probably a good measure for the number of cells on earth. Even if plants manage to come to that level, the order of magnitude will probably not change (by much).