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Why Can’t I Hold All These Limes? is an image macro series featuring a man attempting to carry several limes in his hands while they appear to be falling from his grasp. Derivatives often feature other characters failing to secure a variety of objects.
The image of the man holding the limes comes from a photo of a display ad for George clothing, perhaps from a Walmart.
George is the name of a clothing brand name owned by the British supermarket ASDA (ASquith DAiries), who are owned by Wal-Mart. The reason he is holding limes is to look friendly and also because ASDA has the colour green traditionally. 
KYM User Morgan Lowe made the following statement regarding the origin:
Wow, On april 12th of 2010, My family and I were at walmart. I saw the ad and thought wow how stupid… Limes? Of all things LIMES?!?! LOL I tooke a pic with my phone and uploaded it to facebook. I guess other people did the same haha Never would have thought it would be a meme! I flipped out when I saw this because of all the laughs about it haha good times.
The first image macros are suspected to have come from 4chan, but there are no archived threads to confirm this.
The image macro with the overlaid text “Why can’t I hold all these limes?” was posted to Funny Junk  on July 24th, 2010. A Green Oval thread  in the Anime / Mango board was created on August 25th, and featured several limes guy derivatives. A Yahoo! Answers question  was created on September 2nd, 2010. A Reddit thread  was posted on November 13th, 2010 with a photo of the display ad. In February 2011 a Canv.as thread was started.
Reddit – Meet George, the man who will never be able to hold all those limes :( / 11/13/2010