I have not heard of any such sites, but I would love for them to exist. Since most of my writing is poetry (which is not up to par compared to the poetry that I read and listen to), I would love to find things that put forth those challenges.
Right now, all I can find for those types of challenges are in books. They are primarily how-to books with input from various writing professors and teachers. For example, Rita Dove is a professor at the University of Virginia. She is also a Pulitzer Prize winner, former U.S. Poet Laureate and former Poet Laureate for Virginia. One of her lessons was to write any type of poem that would contain 12 words that she provided. For another type of challenge, I attempted to write a sestina poem with the same six repeated words that Elizabeth Bishop used in her poem "Sestina".
I do try to challenge myself to write at least 1000 words everyday. I try to stay realistic in that I do not expect those words to go toward one specific piece. If they do, that's great, but if not, there is always brainstorming for the next writing project.
I have also asked friends and family to put forth certain challenges for me. Unfortunately, they "don't have the time", or since they are not writers, they do not understand what I am looking for. The most frequent advice that I get from them is "just write whatever you want" or "write something 'happy' for a change". - "Uh ... yeah .... great .... thanks for your help .... NOT!!!!"
I do know that, lately, I have not been very focused. So, that is my latest challenge to myself - to get focused. Otherwise, HubPages will just be a blogging site like any other, which means "not paying".