Upcycling Ceramics

Broken ceramic anything, broken glass anything

Raw material for mosaic artists. Such an old art form it's been found in the ruins of ancient Rome and Pompeii. If you like doing this kind of art, putting an ad out online can help you stay inundated with free raw materials from neighbors for years. Here are some examples at DIYnCrafts but there are legions of other ways to put mosaic work to use in daily life. It's a very versatile craft.
Mosaic work adds interest to some concrete work.

Scratched up plates and saucers

Use either as candleholders or as relatively heat resistant trivets for other candleholders atop those. My family's been using these for years, for instance to hold up a pillar candle on a Wiccan altar (that would be mine), or to affix candles for a chanukiah for Hanukah (that would be my family's). The trick with the latter is to use a match or lighter to melt the wax a little on the opposite side of the wick then use that melted wax to stick it on the plate. BTW if you have neither of these aluminum foil atop a cutting board or baking tray will also work.
These work as saucers for outdoor potted plants. If they don't leak, you can use them for indoor ones too
Smaller plates help define spaces and work as catchall trays for things like keys, coins, and other doodads that would otherwise lay scattered on a side table or kitchen table. Or, really anywhere that you have collections of items that are best stored loose and in sight but are both a. small and b. have a tendency to wander off

Chipped teacups or mugs

Chipped mugs

Chipped and/or scratched up bowls

Can be the top part of a birdbath
Use these to display potpourri, especially if you can cover the bowls with lids or plates when not required to scent a room and be on display
Can be sunk into the ground to help provide sources of water for wildlife just a little bit after rains. Add a little bit of gravel to make a slope from one side to the other and see what shows up. It's not ideal, since its water won't last super long after rain, and it's not huge, but it's something.
Put one on a patio to catch rain. See what shows up to drink. Might help bees. Or wasps. Add a few rocks in there if you can for them to land on. If you like doing this, then after that evaporates, you can replenish it with water from the tap or something.
Put to use as little catch-alls in each and every room of the house that needs one, such as in each bedroom, especially for holding car keys for each person that needs a place to stash them, also wallets, cell phones, and other crucial pocket items that have a tendency to get lost

Old and lead- and other contaminant-free bathtub

Put it in a shed. Soak basketry materials in it. Or, make it into an oxalic acid bath tank for finished baskets.
Sink it into the ground a good foot or so, fill it up, and install a wildlife pond in there, using good practices to ensure it's a healthy pond instead of stagnant. Directions for this can be found in Creating Small Habitats For Wildlife In Your Garden by Josie Briggs. You can also rig up a recirculating waterfall in there or leading to there, using one of the many tutorials online, a beefy robust water pump connected to an outdoor GFCI outlet by a licensed electrician, and a good filtration system so the pump doesn't get clogged
Do the same thing with it, but where water normally collects after rainstorms under a gutter's downspout to make a rain pond garden. You'll need to ensure your roof and gutter system don't collect contaminants and funnel them into the runoff by looking up the materials safety data sheet for said materials and their coatings, and chatting up helpful folks at hardware stores and on builder's forums and plumber's forums online about this, so this isn't suitable for everyone.

Old and lead- and other contaminant-free sink

Take out the plug and cover it with a wad of cloth or mesh or a white cotton rag or washcloth to provide drainage without letting dirt fall out of the hole. Walk it outside. Fill with dirt. Plant stuff in there. Top with rocks, straw, or cedar mulch to conserve moisture if needed.

Old but intact saucers

Use as soap dishes, or to put a spoon on in the middle of cooking, or to put assorted drippy stuff on in bathrooms and laundry rooms and kitchens

Crappy crumbling bricks

Still fine for using to make edging for garden paths, and to construct raised beds with. I very much suggest still using the best techniques for bricklaying that you can, which are well explained in Brick Projects for the Landscape by Alan and Gill Bridgewater and Building with Masonry by Dick Kreh