Well it's getting more and more difficult to find any proven characters, but I'll throw in a few "reasonable" possibilities, at least trying to cover 6 out of the 7 episodes.
Supreme Leader Snoke
This villain from Episode 7 looks very old, if he is more than 62 years old then it means he was alive in Episode I.
Nien Numb is Lando's copilot. We can see him in Episodes VI, and 30 years later in Episode VII. If he was the same age as Lando, or older, he might have been alive in Episode I as well (see Lando below).
Lando is older than Han Solo, and according this webpage was born in 31 BBY. I'm not sure how the webpage found that out. It might be possible that he was born during Episode 1.
He didn't die in the movies and appeared in many games of the extended universe after Episode 6, so he might still be alive in later episodes.
Jabba's servant can be seen in Episode I and Episode VI. He does not die in episode VI. He might be still alive in Episode VII.
Chirpa the Chief Ewok
According to canon information, he was the chief of Bright Tree Village on Endor for 42 seasons, which mean, on Endor's 402 orbital period, approximately 12 standard years, which means since 8 BBY.
If he was born after episode I (32 BBY), it means he became chief of his village at only 24 years old, which is possible but seems a bit young, when also considering the fact that after Episode VI he retires and leaves his place to his daughter Kneesaa. It would be a bit too young to retired before 36 years old wouldn't it ?
The Exogorth from Episode V
The gigantic asteroid worm who nearly ate the Millennium Falcon in Episode sure looks more than 40 years old. He looks quite badass and I don't see why someone would explode his asteroid so he must have survived until the next saga ;)