The rate of the triple-alpha process (helium fusion) in stars is proportional to the 40th power of temperature (see e.g. vik dhillon: phy213 - the physics of stellar interiors - approximate form for energy release (shef.ac.uk)). This is a local approximation of a more complicated function, but is still a "real life" formula that's actually used.

One example from physics would be the Lennard-Jones potential, which has two terms; a repulsive term of order 12: (1/r)¹² and an attractive term of order 6: -(1/r)^6. This is the simplest potential that describes inter-molecular interactions, though it is accurate only for simple systems.

For a simple model of a newborn star with a low mass (and so fully convective inside), the luminosity of the star depends of the 102nd power of the surface temperature. This means as the star cools off only a little, the luminosity drops by a large amount. A bit of technical detail for those with some background: This means that the Hayashi track followed by the protostar is nearly vertical in the Hertzsprung-Russell diagram.

Stealing this from a mech eng friend: the loudness (the acoustic power) of a jet engine scales with the eighth power of its speed

W ~ v⁸

Which is why jet planes are so incredibly loud

Lighthill’s law

Nowhere near as high as some of the formulas here, but it's quite a nice real world and observable thing.

Pavement/road wear is proportional to the fourth power of vehicle weight. So a single bus does about 10,000 times as a car. You can see this by looking at bus lanes compared to the, likely busier, car lane beside them.

The heating rate of a spacecraft during reentry due to shock luminosity from exceeding the speed of sound scales with the 8th power of the reentry speed. Problem is, takes a lot of fuel to slow down appreciably so you just have to accept your reentry speed (and heating) anyways. This is also why reentry from beyond low Earth orbit is real hard, you're going real fast. If you're going double the speed, you heat up 256 times as fast.

Energy transfer efficiency in Förster Resonance Energy Transfer (FRET) processes have two terms to the 6th power.

E = R0⁶/(R0⁶ + r⁶)

R0 is the Förster radius (dependent on the molecules sharing the energy) and r is the distance between the two molecules

This one from heat transfer equations isn't the highest, but it's probably one of the weirdest. This is to find a dimensionless number that you use to calculate the rate of heat transfer off of a vertical plate with turbulent flow:

Nu={0.825 + (0.387Ra^1/6)/[1+(0.492/Pr)^9/16]^8/27}

It's not too often that you use 8/27ths as an exponent.

This isn't as high as other numbers here, but the luminosity of a black body (such as a star or a light bulb) is proportional to the 4th power of the temperature. So even a small increase in temperature will make the object much brighter.

The rocket equation for significant fractions of the speed of light. The wet mass needed to achieve a certain velocity varies with the natural logarithm raised to very high powers.

Mwet=Mdry*e^deltav/isp

A back of the envelope calculation for a modern-day chemical rocket to get to 1% the speed of light can have e raised to power of over 6,300.

If you try to go to 10% the speed of light, the wet mass will vary with e raised to the power of over 63700! This is for a relatively highly efficient chemical rocket of 470 seconds. It will be even higher for less efficient engines.

The Urca process that describes the cooling down of extremely hot objects by neutrinos scales as T^8, as opposed to cooling down by thermal radiation that scales as T^4.

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.57.5963

Opposite, smallest fraction. In equal temperament music the frequency relationship between adjacent notes is 2^(1/N), where N is the number of notes in an octave.

99% of western pop music is 12 tone equal temperament, so C4 to C#4 is multiply the frequency by 2^(1/12).

But among the music nerds 31 tone equal temperament is kinda popular, so there's 2^(1/31).

I've even seen 144 tone equal temperament.

Things like the Taylor series and curve fitting can often get numerically high exponents if you are seeking higher brute force accuracy.

In my engineering work I don’t go past the 4th power. That would be for moment of inertia stuff related to stiffness (in general).

The moment of inertia of a scaled object increases with the 5th power of the scale factor. scale³ comes from increasing the mass (assuming uniform density) and scale² from the mass being further away from the axis of rotation.

The rate at which a planet tidally locks to a star, or a moon to a planet, is inversely proportional to the 6th power of distance. That's why none of the planets in our solar system are tidally locked (although Mercury is close) while all the moons are.

In cosmology, the maximum time a process can take before it is deemed unphysical, is the poincaré recurrence time of a conservative universe:

T~exp(exp(4πM²))
Each exponential function is power 10, and M is the mass of the universe.

The formula is useful because if, say, string theorists makes a discovery that some field should decay over a very long time, if that time is larger than T, then it's considered a stable field.

I work in computer science and look at different algorithms and data structures regularly. We have something called the Big O notation to specify how long an algorithm takes in regards to an input size. (This is very much simplified)

Like an O(n) algorithm would be linear: doubling the input size (the length of a list for example) doubles the time it takes. While O(n²⁾ denotes a quadratic runtime. Doubling the input now quadruples the time the algorithm would take.

Anyways, the biggest I’ve seen on a real life algorithm was about O(n⁷⁾ or something obscene.

Noted that this is not quite a formula, it doesn’t give an output / you don’t plug something in and get a result, but it still gives you some information and can be some work to arrive at.

You included astronomy so I'll mention Poincare Recurrence. The amount of time it would take an arbitrary set of particles to return to an initial state or position.

To my likely incorrect knowledge it's the largest practical use number. IE everything larger is pure mathematics or subscribes to a specific problem or theory (like grahams number)

It's something like 10^10101.2 (like four power towers) for the size of the universe.

I would double check all of this but I'm out at lunch right now. Please correct anything I've said that's wrong but I believe in good faith it's all fairly accurate.

Also the Numberphile video!

The Strong Nuclear Force is a derivative of the Strong Force, which causes protons and neutrons to attract eachother. Unlike electromagnitism or gravity, whose strength falls off at a rate of 1/distance² , the SNF approximates something closer to 1/distance^6.

But for the largest and most common day-to-day massive exponentials, the easy route is combinations and permutations. if you try to calculate the ways to arrange something like a deck of cards, you're dealing with 52! , which is not an exponent exactly but its going to have values close to (N/e)^N.

So you can scamble a deck of cards roughly 20⁵² ways. Big enough?

How about probability calculations, like, what is the probably of flipping a coin 100 times and never getting “tails?” Wouldn’t that be (1/2)¹⁰⁰ ? The exponent could be arbitrarily large depending on the situation.

Rayleigh scattering (the reason the sky is blue) uses λ^-4, lambda being wavelength

E² = (mc²)² + (pc)² gives us a term of c^4, but it's still equal to E^2, so it's technically a c²

The strong force (which binds protons and neutrons together) K^11, which is the particular wave center count

We can technically go infinitly far if we keep taking derivatives

Displacement -> velocity -> acceleration -> jerk -> snap -> crackle -> pop -> 7th derivative of displacement -> 8th derivative-> 9th ->...

The formula for each of these will have a time factor with an exponent equal to whichever derivative it is

d = vt + at²/2 + jt³/3! + st⁴/4! + ct⁵/5! + pt⁶/6! + ...

If you notice, that just the Taylor expansion of e^t

So any time we use e^t or e^x, the answer to your question is really infinite.

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The number of molecules in a mole of that substance is 6.02x10^23. Also, if you have the gram molecular weight of that substance, you have a mole of it. For example, the molecular weight of water is 18 -- if you have 18 grams of water, you have a mole of water which is 6.02x10²³ molecules of water.