Most common, worth buying

Mat
Resistance bands
Jump rope
Bosu ball (does not have to brandname)
Some weights

Nice to have

Dumb bells: 1-2 sets in different weights
Kettlebells : 1-3 in different weights
Swiss ball
Box or Bench
TRX
Foam roller

Items to substitute from around the house

Chair or bench
Stairs or step
Heavy objects like soup cans or full water bottle
Backpack (to fill with heavy objects)
Towel
Book or something else to jump over