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I have four kids. YMMV, but I've found with mine that six years old seems to be the magic threshold. Developmentally, it seems like at that age a switch suddenly turns on and they start really showing distinct interests, talking fluently, thinking more abstractly, making solid friendships with a few peers, and wanting a bit more independence. I find the 6yo age to be a blast.
I'd echo this. My youngest is 5, and though we went through the terrible 2s through the f**ing 4s, it seems we've turned a corner at age 5. There's this sweet little kid who can start to control his emotions and frequently makes us laugh. In place of fits and tantrums we had for basically the past three years, we have someone who can take a breather and accept "no" for an answer. The best part is all of the hugs, the…
When they're much younger you constantly have to fret over them, but now she's learning to write, asks fun questions and overall the "spark" of being her own person is really visible.
While I used to enjoy babysitting earlier too, back then I still used to be glad to be done so I could rest and catch up on some hobbies while nowadays I usually feel pretty guilty having to leave.
On the other hand the experience from my niece and now my nephew does make me wonder if I really want kids at all given the effort it takes to raise them to the point where they start to learn to be independent. I was exhausted dealing with them for just a few days in a row, I can't imagine how I'd handle it for several years.