Larry - certainly the best known character in the world of similes. Most likely to be an Australian or New Zealand expression.
The earliest printed reference currently known is from the New Zealand writer G. L. Meredith, dating from around 1875:
"We would be as happy as Larry if it were not for the rats".
Almost all the other early citations are from Australia or New Zealand. For example, this from Tom Collins (the pen name of the popular Australian writer Joseph Furphy), in Barrier Truth, 1903:
"Now that the adventure was drawing to an end, I found a peace of mind that all the old fogies on the river couldn't disturb. I was as happy as Larry."
There are two commonly repeated contenders for the derivation. One is that it refers to the Australian boxer Larry Foley (1847 - 1917). Foley was a successful boxer who never lost a fight. He retired at 32 and collected a purse of £1,000 for his final fight. So, we can expect that he was known to be happy with his lot in the 1870s - just when the phrase is first cited.
The alternative explanation is that it relates to the Cornish and later Australian/New Zealand slang term 'larrikin', meaning a rough type or hooligan, i.e. one predisposed to larking about. 'Larrikin' would have been a term that Meredith would have known. The earliest citation of that is also from New Zealand and also around the time of the first citation, in H. W. Harper's Letters from New Zealand, 1868:
"We are beset with larrikins, who lurk about in the darkness and deliver every sort of attack on the walls and roof with stones and sticks."