when does the noun "time" become a countable noun?

In the sentence "I am having a good time", you are not using time in the sense of time on a clock. This sentence is equivalent to saying something like "This moment in time is great.", So that sentence is clearly correct. Also, time is countable in that sentence, because it is specifically "a time.". In the sentence "I don't have the time.", time is not countable. This is because you don't have a set amount of time. It could be any amount, therefore it is uncountable.

"Time" can mean either the passage of time e.g. "time waits for no man", or an occasion or moment e.g. "we had a great time last night" . The passage of time is not countable, but occasions or moments are countable.

Other languages have different words for these which may be less confusing, e.g. in Portuguese, tempo often refers to time as in the passage of time, and altura means time as in an occasion or moment.

Time: When referring to multiple past memories I would say would make it a countable noun.

"My best friend and I share a lot of good times"

"All the times I went to the grocery store during the night, they were sold out of banana's"

Wiktionary gives the following countable usages for time:

A measurement of a quantity of time; a numerical or general indication of a length of progression. a long time; Record the individual times for the processes in each batch. Only your best time is compared with the other competitors. The algorithm runs in O(n2) time.

1898, Winston Churchill, The Celebrity, chapter 1: “I was about to say that I had known the Celebrity from the time he wore kilts. But I see I will have to amend that, because he was not a celebrity then, nor, indeed, did he achieve fame until some time after I left New York for the West.”

1938, Richard Hughes, In Hazard: “The shock of the water, of course, woke him, and he swam for quite a time.”

An experience. We had a wonderful time at the party.

1898, Winston Churchill, The Celebrity: “I was about to say that I had known the Celebrity from the time he wore kilts. But I see I will have to amend that, because he was not a celebrity then, nor, indeed, did he achieve fame until some time after I left New York for the West.”

An era; (with the, sometimes in plural) the current era, the current state of affairs. Roman times; the time of the dinosaurs

63 BC, Cicero, “First Oration against Catiline” (in translation): “O the times, O the customs!” (Originally, “O tempora o mores”)

1601, William Shakespeare, The Tragedy of Hamlet, Prince of Denmark: “The time is out of joint”