How many more will be made?

The existence of the Enterprise implies the existence of Backspaceprise

At least four or five according to on screen information.

I believe the meme above refers to:

1. Archer's NX Class (NX-01)
2. Kirk's Constitution class (NCC-1701 with no suffix, heavily modified to Constitution refit class for feature films 1 through 3)
3. Kirk's Constitution refit class (started the letter suffix system with A)
4. The Excelsior class seen briefly in Generations
5. the Ambassador class seen briefly in an episode of TNG
6. Picard's Galaxy class
7. Picard's Sovereign class (the most recent in the series with a suffix of E)

As the seven starships named Enterprise.

There is an 8th that has been seen on screen; Archer is given a brief glimpse of the future Enterprise J by his fairy godmother Crewman Daniels. This implies that an F, G, H, and possibly I will be commissioned. I is sometimes skipped with registration numbers because it can be confused with the number 1. So if the show's in-universe "present day" is where ST: Picard is, a few decades after the events of TNG, there are four, or possibly five, Enterprises implied or confirmed in canon to be built over the next ~250 years.

They had to make sure history never forgot the name... Enterprise.

Actually that answer implies some nerdy knowledge.