My oldest son makes custom Lego sets, and he ships the sets using the Post Office’s Priority Mail service. On the off chance that you’ve never used it before, Priority Mail uses four different standard boxes, each with a different size and shape.

Not surprisingly, the four boxes have different shipping prices, with the smallest one the cheapest at $10 and the largest one the most expensive at $25. But perhaps more interestingly, the four boxes all have the same maximum weight: 70 pounds.

It’s not hard to imagine putting 70 pounds of stuff in the largest box. At 12 inches by 12 inches by 6 inches, its volume is half a cubic foot. Filled with concrete, it would weigh about 75 pounds and would place you squarely in violation of the Priority Mail rules and regulations.

But what about the smallest box? With an interior of just 8 5/8 by 5 3/8 by 1 5/8 inches, the smallest box is pretty tiny, about the size of “three stacked DVD cases” as the post office website helpfully suggests. How heavy could “three DVD cases” be? Is there any danger of exceeding the 70 pound limit?

To find out, let’s first dispense with the inches and pounds and move to the metric system. The tiny box’s interior dimensions work out to about 21.9 by 13.7 by 4.1 centimeters, for a total volume of about 1200 cubic centimeters. Our max weight of 70 pounds is about 32 kilograms. Is there anything we can get our hands on that’s so dense that it’ll make our tiny box weigh more than 32 kilograms?

Concrete is right out. While it was dense enough to make the large box break the weight limit, concrete is only about 2.4 grams per cubic centimeter, so filling our small box with concrete would make it only weigh about 3 kilograms. That’s barely one tenth of the weight we need to break the limit! Clearly we need something much more dense…

What about lead? Lead’s density is indeed prodigious, inspiring English idioms like “lead foot,” “get the lead out,” and “going over like a lead balloon.” But even at 11 grams per cubic centimeter, our lead-filled small box would weigh “only” about 14 kilograms or 31 pounds. We’re not even halfway to the maximum weight!

Mercury is only a little better. With a density of about 14 grams per cubic centimeter, a mercury-filled box would weigh about 17 kilograms. So a mercury-filled box would put you about halfway to the max weight (and potentially put you in jail, since toxic mercury is illegal to mail).

At a whopping 19 grams per cubic centimeter, Uranium is extremely dense. Depleted uranium is used for armor-piercing ammunition in part because its enormous mass carries more kinetic energy than conventional ammunition. And a uranium-filled small Priority Mail box would weigh about 23 kilograms, or about the average weight of a 7-year-old child. But we’ve still got 9 kilograms to go before we reach the maxiumum weight.

Probably the most dense substance that normal people can own is tungsten, a metal whose name means “heavy stone” in Swedish. I have a one-inch cube of tungsten on my desk, obtained for about $100 from Midwest Tungsten Service, and it’s always fun to drop it into the hand of someone who has never held tungsten and see their shocked reaction. It’s uncannily heavy, so heavy for its size that it’s almost like an another force in addition to gravity is trying to drag it down to the floor. Tungsten is 19.3 grams per cubic centimeter—almost twice as dense as lead. Everything more dense than tungsten is either extraordinarily expensive or not something that you’d want to be in the same room with. But if we fill our little box with tungsten, it’ll tip the scales at “just” 24 kilograms. Still not too heavy to mail.

Osmium is the densest naturally-occurring element at 22.6 grams per cubic centimeter. And this rare platinum-group metal has a price to match: about $1.4 million USD per kilogram as of 2024. Filling our box with osmium would require almost 28 kilograms of osmium, which would set us back a cool $40 million USD, which is supposedly the insurance value of Taylor Swift’s legs. And our osmium-filled box still wouldn’t exceed the weight limit for the small Priority Mail box!!!

So that’s the answer to my question. Is there anything that you could put in a small Priority Mail box that would exceed the maximum weight of 70 pounds? No. If you want to use a small Priority Mail box to send someone a Taylor-Swift-legs-worth of osmium, go right ahead, the Postal Service won’t argue with you.

Well, maybe you can exceed the weight limit of the small Priority Mail box, but you’ll have to leave the solar system to do it. The closest neutron star is about 400 light years away, and it has a density of about 3.7 x 10^20 grams per cubic centimeter. A small Priority Mail box filled with neutron star matter would weigh about 4.6 x 10^20 kilograms, or about the same as Tethys, Saturn’s 660-mile-wide moon. And that would most definitely exceed the 70-pound weight limit for Priority Mail.

So if you’re mailing a neutron star, use FedEx.