The 34-year-old will hang up his boots following his 431st appearance for the club against Gloucester at Sandy Park in the Aviva Premiership on Saturday.
Baxter believes that he and his younger brother have been on an incredible journey together at the club.
"It's been an incredible story, the amount of good rugby he's played and he's played all his rugby at Exeter when we've been playing at our highest level, either through the Championship, Premiership or Heineken Cup," coach Baxter told the club's official site.
"The longevity of what he's done has been fantastic, the loyalty he's shown has been fantastic, but I just think it's fantastic that someone like that had the opportunity to play for Exeter in the Premiership and Heineken Cup because no one's deserved it more.
"He's been a huge part of how the club got to where it is now and it's going to feel a bit strange for me personally after Saturday not having him around."
Richard made his Chiefs debut 16 years ago in October 1997.