Monger Phone
 

Is there an app for your cell phone to make and receive calls and texts and the other person’s phone number is not stored into your cell phone’s call log history on your monthly invoices?

Thug, you can use wechat or whatsapp for calls if the other party has those apps. otherwise, you need a burner phone, dude.

Textnow will allow you to make phone calls and texts.

Burner is the ABSOLUTE best option! TRUST ME! (Except my Rep Power has actually gone down. What's with that? But I digress.)

Texting/Calling Apps
 

Check your app site (Apple/Google) for several online phone apps with phone numbers. Most allow free texting and making and receiving voice calls with the remittance of cash or other compensation.

With a burner phone, you dont have to remember to do anything but leave it hidden in the car with the ringer off!!!!!

Check out "Burner" as NTK has recommended. Works great with IPhone.

Tmobile
 

I've been using TMobile for quite a number of years, now the plan I have is one of their grandfathered plans.
I can buy minutes, $100, get their Gold status, which means all minutes are valid for a year. Once the year is up, I can top up for $10, which will extend the expiration of all minutes another year.
Pretty good deal.
Currently, TMobile has this plan
https://prepaid-phones.t-mobile.com/pay-as-you-go $3/mo, which isn't too bad

google voice also does it, been around almost a decade.......

Google voice integrates differently with different carriers and versions of Android. On some carriers, it doesn't meed the OPs criteria: to no show the call on the "cell phone’s call log history".

Depending on how you configure it, and the version you're using, it'll merge Google Voice Text with yoru regular texts, and calls with regular calls. On some carriers it'll even show up on the monthly bill.

Textnow is completely independent of the carrier and phone's phone call and sms log. It has the added advantage that if you don't use it for 2 weeks, they'll recycle your phone # and you get to select a brand new one, so that the mamasan's can't keep tabs on you and thinks you're a new caller/texter every time. Wiht Google Voice it's very hard to get a new phone # after you've been assigned one.