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Unfortunately, I believe it is technically illegal to record a phone call without informing the person on the other end of the line...so I don't think any mobile phones do this.

However, you can buy special recording devices that hook into a wired phone line. If you've still got analog phone lines (no VoIP) in the house, you can try that.

that's not true actually...most states have whats called "one party consent" laws which basically states that at least one of the callers has to give consent to record the call...check your state law...Google Voice records calls...just press 4 while on a call
 
If you use google voice, you can record any incoming call made to your GV number, with any phone. Also you can block any number or you can screen calls too.
 
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