The line here is a bit arbitrary, but sometimes you want a plant that's going to just lay there, and sometimes you want a plant that's going to climb up things. This list is for the plants that are just going to lay there. A lot of these are used as hanging-basket plants.
A few of these plants will climb to some degree if they're being grown outside (some, like Ficus pumila, are downright obnoxious about it), but indoors, you generally have to provide them something to climb and then anchor the stems with some kind of plant tie at regular intervals, or make them some kind of constantly-moist support, or some other impractical thing like that. Most of the plants in this list won't climb particularly well even with assistance, and can be relied on not to get any crazy ideas about running themselves up your curtains. Your mileage may vary, of course.
This is an even less comprehensive list than usual, because the number of plants with this basic growth habit is practically never-ending. I'll add suggestions as people are moved to leave them in comments.
Note: this list was done by request for Joseph Tychonievitch of Greensparrow Gardens, who asked. I don't promise to take all list requests (some lists just can't be done as easily as others), and anybody who wants one is going to have to take a number and wait anyway, because Joseph asked for FOUR of them, but I'm more or less open to doing posts by request, in theory. Just so you're aware.
EDIT: Whoops. I published this without adding any commentary on the pictured plants.
I guess my three favorites from the above list are probably Aeschynanthus spp., Gynura aurantiaca, and Pellionia pulchra, with Cyanotis kewensis serving as first alternate.
This is partially process of elimination, because I've had moderately bad experiences with a lot of the other plants on the list, either from bugs (Maranta, Hedera), rot (Epipremnum), or decline from unknown causes (Pilea, Senecio). And I've never actually owned a Dischidia.
Callisia fragrans (basket plant)
Ceropegia woodii (string of hearts)
Chlorophytum comosum (spider plant, airplane plant, mala madre)
Cissus rhombifolia (grape ivy; oakleaf ivy) (will climb a little)
Cissus rotundifolia (Peruvian grape ivy) (will climb a little)
Crassula muscosa (= C. lycopodioides) (watch-chain crassula)
Davallia spp. (rabbit's-foot ferns)
Dischidia nummularia ("button orchid," says davesgarden.com, though it's not an orchid)
Episcia cvv. (flame violet)
Ficus pumila (climbing fig) (will climb a little)
Fittonia albivenis (nerve plant)
Hedera helix (English ivy) (will climb a little)
Hoya carnosa (will climb a little)
Hoya lacunosa (will climb a little)
Hoya kentiana (will climb a little)
Hoya kerrii (sweetheart hoya) (will climb a little)
Hoya pubicalyx (will climb a little)
Ludisia discolor (jewel orchid) (Also other "jewel orchids" like Goodyera, Anoectochilus, Macodes, etc., though those are a lot harder to find.)
Monstera adansonii (swiss cheese philodendron) (will climb a little)
Monstera deliciosa (split-leaf philodendron, swiss cheese philodendron) (will climb a little)
Nematanthus gregarius (guppy plant)
Philodendron bipennifolium (horse-head philodendron) (will climb a little)
Philodendron erubescens 'Red Emerald' (will climb a little)
Philodendron hastatum (will climb a little)
Philodendron hederaceum (heart-leaf philodendron) (will climb a little)
Phlebodium aureum 'Mandianum' (bear's paw fern)
Pilea nummulariifolia (creeping charlie)
Plectranthus verticillatus (Swedish ivy)
Polypodium formosanum 'Cristatum' (E.T. fern)
Rhipsalis spp. (mistletoe cactus)
Saxifraga stolonifera (strawberry begonia)
Schlumbergera truncata cvv. (holiday cactus, Thanksgiving cactus, Christmas cactus)
Scindapsus pictus (satin pothos) (will climb a little)
Scyphularia pycnocarpa (possum tail fern)
Sedum burrito (burro's tail)
Sedum morganianum (also burro's tail)
Senecio jacobsenii (trailing jade)
Senecio radicans (string of bananas)
Soleirolia soleirolii (baby tears)
A few cvv. of Solenostemon scutellarioides (coleus)
Stapelia spp. (carrion flower)
Stephanotis floribunda (Madagascar jasmine) (will climb a little)
Streptocarpus saxorum cvv. (streptocarpella)
Syngonium podophyllum (arrowhead vine) (will climb a little)
Tolmiea menziesii (piggyback plant)
Tradescantia pallida (purple heart, purple queen)
Tradescantia sillamontana (kitten ears)
Tradescantia zebrina (wandering Jew)