Melissa, I don’t think it’s a word missing from the English language.  In fact, I think the word you’re looking for is “game”.  If I were to say “I played 6 games of Chess last weekend”, you would have no trouble understanding me.

The problem is that the single word “game” has two different uses (it’s “overloaded”, as some people might say.) As a result, it’s hard to understand statements in which either use is possibly right.

I’ve used the term “playing” where you’re using “play”, but I agree that it would be very helpful to coin or co-opt a new word to fill this gap in our vocabulary and take some of the load off “game”.

Why not use the German/French word “Partie”?
After all, they’ve already borrowed enough from our language, it’s only fair to return the favor.
Besides, it sounds like “Party”!

Semantics are equally frustrating and fascinating, to me.  As an academic, I often deal with students that get tripped up by words with double meaning--a technical meaning in the discipline but a different meaning in common use (for example, the term correlation is often used/misused by the general public to simply mean relationship, but in statistics it only refers to a linear relationship).  I also struggle with an ever-changing vocabulary as required by political correctness.  For example, the most recent edition of the APA required that we switch from the term Subjects to Participants.

I guess I generally use “X plays of N games” rather than “X playings of N games” or “X games of N games.”

That said, I still don’t see the difference between “In this game, the starting bell...” and “In this play, the starting bell....”

Thanks for the comments.
Eric, you’re quite right that it’s an issue of one word with two meanings rather than a completely missing word.
Jeff, I suspect that the similarity to Party is what kills that suggestion cold. “Let’s have an Ra party” makes me think of bad egyptian costumes rather than auctions.
Valerie, to me the difference in those statements is that the first - using game - has the weight of generalisation. The Starting Player ALWAYS has the bell however you play - whereas in the other case there may be some condition affecting this *play* that gives the bell to the starting player.
It’s a contrived example, I know.

To complicate your issue play is also overloaded.  It can refer to a single actions is in saying ‘nice play’ when someone hits the bullseye on a dartboard.

Would be nice to have a new stand-alone term that has no other meaning, but this being the English language how likely is it?

Even if we were to go with a new word, it is likely to be used in some other way later.

I would like to propose the word Zong, as in 38 zongs of 18 games.