Would it be possible for a chemical mixture to explode as a result of not being shaken?
Rather than the ‘be careful with that’ test tube of nitroglycerin, is there some kind of compound that will explode if left still?
Nitroglycerin is unusual among modern explosives in being so shock-sensitive. (Dynamite, in case you didn’t know, is nitroglycerin embedded in a stabilizing clay matrix. Nobel got rich making nitro reasonably safe to handle.) Most modern explosives can be burned, dropped, stomped on etc. - only the very sharp impact of a detonator cap will set them off. (Do NOT stomp on the next brick of C4 you see; it’s not THAT safe.)
For any explosive to be detonated, there has to be a triggering action, be it chemical, electrical or physical. Picric acid is notorious for aging into crystals on the shelf, which can be detonated with the slightest bump. (When I was in high school, there was a district-wide call to locate old jars of picric acid in the store rooms, and not a few had to be removed by bomb squads.) In theory, one of the really touchy explosives like picric acid crystals, mercuric chloride (?) etc. could self-detonate from a bump, vibration, static discharge or a heat rise.
But SOMETHING has to set it off, even it it’s chemical change over time.
(Added, re first sentence: there are certainly many chemical compounds that are shock-sensitive, but they aren’t usually used as explosives.)
Some children will explode if forced to remain still for too long.
How about nitrate-rich fertilizer or even a compost pile? Doesn’t a large pile of the stuff have to be regularly turned to keep it from spontaneously combusting?
Minor nitpick to an otherwise excellent response; C4 cannot be initiated by any normal mechanical shock such as dropping it, shooting, et cetera. A high energy pyrotechnic shock (which both heats and creates a shock wave in the material) is required to initiate detonation.
There are substances that will break down over a period of time and spontaneously ignite and deflagrate (burn), but I don’t know of any substance that will transition directly to detonation without an initiating shock or very high pressure. Some energetic substances in quantity can go from burning to detonation (called a deflagation-to-detonation transistion, or DDT) but this requires that the material already be both burning very energetically and under high internal pressure (often caused by local discontinuities like cracks in a solid material, or pressure building up underneath a mass of energetic material).
Stranger
Nitrogen Tri iodine (sp?) may not technically “detonate” but its good enough for gubment work.
I was thinking of a combination of substances - rock or concrete, flint chip, other chemicals present - might be just enough to form a detonator under the right circumstances. If you’re convinced that’s not possible, by all means jump away. I’ll be way over there watching. 
But IIRC, C4 was formulated specifically to be highly safe to transport, even under rough conditions, so…
Like billfish678 I thought of Nitrogen Triiodide; as I recall reading of some demonstrations where crystals of the stuff were detonated with nothing more than the brush of a feather. However, chlorine azide sounds even a bit nastier.
remember, kids, nitrogen really really really really wants to be a gas, and will do any number of irrationally loud and surprising things to get back to that state.
Need answer fast?
Nitrogen tri-iodide will detonate and becomes sensitive to even very modest shock or friction once the excess the excess aqueous solution of amonia evaporates. However, it still requires some kind of outside impulse though it can be as little as friction due to contraction of the drying solution if there is a large enough mass of crystals. (This is why it should only be produced in very small quantities and in situ of the application.) Most explosive substances, and especially those used for any practical military or demolition purpose, however, are not nearly that sensitive due to the obvious hazards.
I guarantee that it will not detonate without anything less than a high frequency, high impulse pyroshock. C4 has been used as fuel for fires, dropped from great height, and otherwise subject to gross mishandling without any recorded instance of unintended detonation in a non-initiated condition. This isn’t a reason to be cavilier around explosives, which should always be transported, handled, and stored in accordance with DoT/DoE/ATFE regulations (e.g. stored in a bunker QD-rated for the appropriate explosive weight) but practically speaking C4 is about as likely to explode as Silly Putty, and becomes less sensitive with age.
Stranger
I’m thinking of small balls of plutonium suspended in a thick oil in a sealed spherical container. Keep shaking it and the balls stay well distributed around the container. Leave it still and the balls all collect at the bottom and reach critical density.
The question of the o.p. was
Would it be possible for a chemical mixture to explode as a result of not being shaken?
Stranger
Can’t be letting the discussion get interesting.
Speed already had a bad sequel. This one sounds even worse.
Just-baled hay, when it is within a certain range of moisture content, and is stacked tightly, will heat up as the hay ‘cures’, and it can get hot enough for the bales to burst into flames. Any farm kid knows this. That’s why you rake hay and let it dry for a couple days before baling it, and why you stack it loosely after baling.
But that’s combustion, not really an explosion.
What about some substance that would naturally spontaneously combust? But say it’s half that, half water. So as long as you are shaking it, the water splits it up enough so that it never combusts… but let it sit still long enough, it might separate, float to the top, and then combust?
There are fluids that are more viscous when shaken. I’d guess you could get some mechanism out of this (e.g., prevent the fluid from mixing properly, or some separation of volatile elements such as in drewtwo99’s suggestion). But that is a mere guess, and I am not sure what counts as keeping to the spirit of the OP.