how many games can you get or download on the nintendo switch right out of the box with no sd card expansion memory?

The ultimate answer is "IT DEPENDS"...

Your Switch has around 26 GB's of usable storage space for Save Data, Snapshots, Video Clips & Digital Games (including updates & DLC). Out of all that content, only Sava Data MUST be stored on the Switch itself.

For the Digital Copies of Switch games, sizes vary between 22.8 MB's (~0.023 GB's) to ~32 GB's... with Save data varying between 0.4 MB's (less than 0.001 GB) to 706 MB's (~0.706 GB), based on some quick research & personal experience.

Therefore, there's some cases where you cannot even fully down load a single game onto your Switch without a microSDXC card. ON AVERAGE, you'll be able to fit a handful of games (around 5) at most before running into storage issues.

I will note that microSDXC card prices are consistently dropping & you can get 128 GB card for around $25. Beyond that, prices jump considerable higher (200 GB's are around $40. 256 GB's around $65) & sizes beyond 256 GB's tend to become price-prohibitive if you stick to retailers. If you're shopping online, you might be able to get 400 GB's or 512 GB's before hitting similar price limits.

Hope this sheds some light on the subject.

Like around 1 or 2 depending on the size I know if you get the cartridge around 3 or 4(extra data) but if you download about 1 or 2 depending on the size

Depends on the game. Some games won't even fit on the Switch without an SD card.

Some games are small enough that you can fit 20-30 games.

SD cards are cheap now. If you are paying $300 for a device, you should easily be able to throw another $20 for a 128 GB card or if you are planning on going all digital, you can get a 400 GB card for under $70 now (I bought mine for $80 a few months ago).

https://amzn.to/2KRMI9w