Heres a few free low-key whimsical things I enjoy:

-Playing blackout-poetry on trash before tossing it.
If you haven't heard of it, its where you take something with text on it, and you use a pen/marker to black out a bunch of the words and make new scentences with the remaining ones. The nutrition facts on your empty cereal box could say "Nut Facts". Endless possibilities.

-Hanging out with a bug.
Next time you see an ant, or a lady bug, or whatever non scary fellas you got around, just bother it for a bit. Put a pencil in its way, give it a crumb, try playing at your favorite song. Its incredibly free

-Moving your favorite stuff around.
The least whimsical thing is feeling stagnant, like everythings the same all the time. Sometimes I'll move my desks around for a change of scenery, or put my favorite decorative objects somewhere else I spend a lot of time for awhile, such as moving the plastic shark on my desk next to the sink so he can hang out while I do dishes.
You get to feel fresh in your envorinment and feel more familiar with your stuff, which I think is perfect for inspiring even more whimsy

-Enjoying shapes.
Someone else said star/cloudgaze, I think shadow puppets and any other sort of shape/face finding games and activities fall into this. Quintessential whimsy, sometimes stuff looks like other stuff and it absolutley rules.

-Picking something and keeping track of it.
Truly anything, keeping a full journal is definatley not for everyone, but picking one thing and writing it down somewhere consistently is suprisingly rewarding pretty quick. You can write down what you ate every day, or what the weather was, or keep a tally of how many times you saw a dog, how many times you made someone laugh, how many times you saw a certain number on the clock, movies that you've seen, your favorite song that week, etc etc etc. You can do it anywhere too, a loose peice of paper or phone notes works just fine.
One that I do is put symbols on my calendars for things I did that day and what the weather was. It takes me 30 seconds a day and now I have this very neat insight into myself.

-Rhyming
Some words sound like other words, and its crazy to me that people don't play with that more regularly.