Small flat rate box, 21.9 x 13.7 x 4.12 cm = 1236 cm^3 = 14kg; you can fill 100% with lead and mail it. Tungsten is also allegedly fine, but it will weigh 23 kilos and be quite difficult to pick up (can't get a finger under an edge...)
Medium flat rate box = 95kg; you can fill it 33% with lead, or ~45% with steel and mail it.
Large flat rate box = 144kg; only 22% lead.
For solids at room temperature and pressure the best you could do seems to be osmium or iridium, unless you have access to heavy transactinides.
mass/kg substance
2.57e5 small flat rate box
2.26e5 osmium
2.25e5 iridium
2.65e5 meitnerium (theoretical)
Considering the expense of synthesizing meitnerium and the half-life which is measured in seconds, I would recommend getting insurance as well as express shipping if you do try.[0] https://en.wikipedia.org/wiki/Meitnerium#Stability_and_half-...
If you've got a particularly slothful postal worker you might consider topping up your meitnerium with a similarly sized sample of roentgenium-282. If you don't have any to hand then perhaps you'll have better luck producing tennessine-294 and waiting a couple of minutes. Amusingly your trip from Tennessee to x-ray land will take you via Moscow and Japan.
(Beware that you might end up making the box bulge a bit, since you'll necessarily go over the size limit by including less dense materials as well as necessary apparatus.)
A buddy of mine used to cast and paint figurines. Well, someone ordered a bunch of lead ones and they used a flat rate to ship it. The box weighed something like 80lbs. It was basically just a block of lead
It's probably coincidence but a few months later a weight limit was placed on flat rate boxes. It's still crazy high. We always thought the timing was funny.