I don't know if this is a stage, or if it's Juliets personality but she is seriously up my ass 24/7.
While I don't mind attending to her needs, she seriously seems unable to be by herself at all. She NEVER plays independently (other than to chase a cat once in a while), if there is something she wants to play with, she'll bring it to me and put it in my lap. If I try and show her how it works and give it back, she'll just give it back to me. I tried stacking blocks with her the other day, and she did ok as long as I was telling her what to do (Ok put another block on top of the first block..) but as soon as I stopped she stopped playing- not even knocking them over or anything. If she wants to draw, she'll bring me paper and crayon and want me to hold the hand holding the crayon and "help". I have a mini slide climbey thing in the house, and she is VERY able to do it by herself, but she wants me as close to her as I can get watching.
My housework is starting to suffer- If I try and fold laundry she gets mad and will pick up and throw anything I have folded, or empty out a draw. I even try and get her to "help" like handing me things, and she won't. Or I get her to help load or unload the dishwasher, and she'll just want to close the door. If I'm cooking, she'll push at my legs to push me away from the stove. She's at the point that she won't even nap unless I'm laying next to her in the bed.
How do I foster just a LITTLE independence (before I go CRAZY)????

    
I'm sorry, I know it's not funny, but that bit is!
Personally I think it's a lot to do with personality, and maybe also first children.  Evie was (is!) exactly the same.  She likes to be no more than 2" away from me at all times.  Henry will play on his own a bit already - probably because he has to because he's never always had my full undivided attention.

It does get a bit better though.  Evie's now old enough to understand that I'll play making things for a little while, then she can carry on while I make dinner etc etc.
She loves helping me with housework (Ok, so it's not helping as you or I know it, but at least she's not whinging while we do it!).  So she'll do dusting or she'll pass me washing to hang up on the line, she likes to fill the washing machine etc etc.  But everything takes a million times longer, and occasionally she'll only want to do the pink things (or whatever).  Cooking - she adores cooking.  Juliet's probably a bit young to really help, but she can have a bowl of dried beans or pasta and do stirring, and pour them into other bowls.  Or sorting out tupperware.  When Evie was younger I'd give her a pile of tupperware and get her to 'sort' it.  She'd make piles with it, put little things into big things, and when she'd 'sorted' it, we'd put it back in the cupboard together, and she was really pleased that she'd done an important job. 

If it's something that I HAVE to do, and I HAVE to do it now - like if I'm making a sauce that needs stirring and she's pulling me away, she'll get a firm 'No.  Mummy needs to do this now, then we'll play'.   Because I think she needs to learn that she can't always have everything she wants immediately.

Maybe generally just start her off playing, and then sit back a bit and try to get her to do it.  Just really gradually remove yourself.

But I don't really know, I don't have any magic answers, which is why I have a clingy child and a messy house!

Jake was and still is that way, while Cade has always been a very independent child. 

I wish I knew what to tell you - but Jake still struggles with this.  What has helped him the most is to have his brother to mimic and play with. 

When I figure it out with all 4 of my kids I will let you know..  I cannot even go to the store without one of the 5 that are living with me now.. 3 of them being adults!!!