Goopey, slippery, stretchy slime is easy to make, and way too much fun.
If you need a proper, dignified justification for mixing up a batch of messy, squishy goop to play with, consider this a lesson in applied material science. Last time, we mixed up flour and oil to create silky sand. This time, the recipe is less edible but way more dynamic.
Your shopping list is borax (a laundry-booster available at drug stores and megamarts) and any PVA glue (wood glue, white school glue). You'll also need water, something to mix things in, and optionally, food colouring.
Mix up a batch of 50/50 water and glue, dissolve a spoonful of Borax in more water, then mix the whole mess together. (If you want real numbers, mix 1/4 c water with 1/4 glue & dissolve 1 tsp Borax in 1/8 c water, but really, you can be pretty slapdash about this.) As you knead it, the slime will quickly start resembling silly putty. For extra awesomeness, consider mixing in some iron filings to create your own batch of magnetic putty.
While I'm all for mixing up weird materials to play with at any age, this is also a relatively cheap and kid-proof toy. You don't want anyone eating this stuff (as Naro eloquently points out, it's essentially a blend of soap and glue), but otherwise it's forgiving to make and, once mixed, not all that messy to play with. You can pre-mix the two solutions for the less dexterous kids to do the final kneading, or the recipe is forgiving of enthusiastic-but-inaccurate measurements as long as the proportions are roughly "A little bit of Borax, a lot of glue, and even more water." Once mixed into slime, you can chuck it into a sealed container to keep it fresh and gooey for later playdates.
The Science of Slime
Image credit: Maki Naro
Viscosity is how resistant a material is to flowing — highly viscous fluids are thick and goopey, while low-viscosity fluids are super-runny. Sometimes the viscosity never changes: water's a good example of something that's the same amount of runniness all the time,* so is an incompressible fluid. But most of the time, viscosity changes when you start mucking about with a fluid. How it changes is the Newtonian/Non-Newtonian divide.
A Newtonian fluid is any fluid where the viscous stress is proportional to the strain. Mathematically, they're linearly proportional, with viscosity as the scaling factor. This makes them the simplest fluid that still incorporates viscosity. A non-Newtonian fluid is anything that's not that. Non-Newtonian fluids have a more complicated relationships with viscosity, frequently even a shear-dependent viscosity.
In nitty-gritty reality, nothing is really a perfect Newtonian fluid, but air and water both act fairly Newtonian without too much hand-waving. You're already familiar with a bunch of non-Newtonian fluids — blood, paint, and shampoo can all get more or less runny in different circumstances. The goopey slime that I'm obsessing over today is viscoplastic: it flows nicely in low-stress environments, but can break and snap like a solid in high-stress environments. It even has elements of elasticity: under the right conditions, you can ball it up and bounce it off the table.
Troubleshooting Faulty Putty
If it's staining surfaces, you added the optional food colouring, didn't you? Limit playtime to non-porous surfaces, disposable surfaces, or accept the marks as part of the stratigraphy of stains giving your table character.
If the end result is gritty, you didn't fully dissolve the Borax into water before mixing. I have no retroactive fix for this, so either tolerate the grittiness or start over.
If the end result is too sticky, you need more Borax. Dissolve more in water, then mix the solution into your putty. Keep adding a little bit more as-needed until you get a good texture.
If the end result isn't elastic, you picked the wrong type of glue. Check the ingredients list to make sure it's a polyvinyl acetate (PVA) glue. In the US, Elmer's white glue isn't, but Elmer's all-purpose glue is. Can't find a PVA glue? Even a non-PVA glue can still be fairly fun to play with, exhibiting the same flow-vs-break dynamics in different stress regimes.
If you want to store the slime, make sure to put it in an airtight container. For extra longevity, stick it in the fridge. You'll need to knead it for a bit to get it warmed up and back into the delightfully slimy-stretchy consistency.
* Glaciers flow but ridiculously slowly, therefore ice has a different viscosity than liquid water yet both can be considered fluids. This is an interesting debate for another day, but is way too far off-topic for DIY slime experiments!