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If I were to try to achieve something you could say I "had a go".

If I tried it multiple times, how would I write that down?

I had many goes

or

I had many go's

or

I had many gos

The dictionaries I've checked seem to be unanimous that the plural of the noun "go" is "goes". I didn't see one list its plural as "gos". However, I didn't check all dictionaries.

American Heritage Dictionary:

n. pl. goes

Collins Dictionary:

n, pl goes

Merriam-Webster Dictionary:

plural goes

Cambridge Dictionary:

plural goes

Random House Unabridged Dictionary (dictionary.com):

plural goes

Google NGram Viewer does not show any results for "two gos at" or "two gos at", but does when written as "goes".

A Google search shows the following results:

"two gos at" = 453 results.

"two goes at" = 19,500 results.

"three gos at" = 252 results.

"three goes at" = 20,200 results.

Note that both Google search and Ngram Viewer may show false positives, as Google search, as far as I know, doesn't take punctuation into account when giving search results, ie., full stops, commas. But I think it's safe to assume the consensus is that the plural of go (attempt or try) is "goes".

I think this is a good question, as somewhere in my head I have an instinct to write it "gos". However the answer is gotten easily by checking some dictionaries.

Also, if you wrote "I had three gos at it before giving up", I have a feeling the typical reader probably wouldn't even be surprised by it or notice it particularly as wrong. This is just my opinion. I have this feeling because the "-os"/"-oes" plural ending rules are wildly inconsistent. Potatoes, tomatoes and heroes are correct. But "photos" and "burritos" are correct. However most dictionaries seem to list either "-os" or "-oes" ending as acceptable in "ghetto" and "mosquito".

Both "mosquito" and "burrito" most likely come from Spanish, yet most dictionaries say only "mosquito" can have plural ending either "-os" or "oes", whereas they're consistent in listing "-os" for burrito plural. Same goes for "canto", "manifesto" and "grotto". These words most likely come from Italian, yet the plural of "canto" is "cantos", whereas the dictionaries say the plural of "grotto" or "manifesto" can end in either "-os" or "-oes". It's actually a tricky topic.

There is an alternate construction which expresses plurality using a singular conjugation, which may be of interest. Tho slightly odd, it has a history of use, and is readily understood:

"many a go"

As in:

We've had many a go at this.

I'll have many a go.

In most usage I have heard, I would consider the phrase "a go at it" to be non-count. In other words, it does not specify the number of tries and thus neither singular nor plural.

Though, VizJS answer does seem to be a logical plural.

Not what the OP intended...

The noun go (a Japanese board game) has the English plural gos. Perhaps one could even say "I had a go" with this meaning.