The only lieutenant governor I remember succeeding to the governorship and being elected more than once is Rick Perry, but Texas has no term limit.
Mike Huckabee succeeded Jim Guy Tucker to the Governorship in Arkansas in 1996 and then was reelected in 1998 and 2002, thus leaving office in January 2007 after 10 1/2 years in office.
But yeah other than him, Rick Perry is the only one I can think of too.
Gary Herbert was the Lieutenant Governor of Utah and became Governor in 2009 when Jon Huntsman resigned to became U.S. Ambassador to China. Utah doesn't have term limits, but Herbert won the 2010 Special Election for Governor, and won reelection to full terms in 2012 and 2016. He's still Governor today (he's not seeking reelection in 2020, though).
Jack Dalrymple was the Lieutenant Governor of North Dakota and became Governor in 2010 when John Hoeven resigned to become a U.S. Senator. He was elected to a full term in 2012. Despite not having term limits, Dalrymple voluntarily decided not to seek a full second term in 2016.