I think most of the stuff we use on site has been indicated - our modest equipment (for a very generic site) includes:
tool boxes and bags
awnings or tents (for shade - with UV 11 and temperatures around 40 degrees you need these here!) - one approach is to use scaffolding with awnings over it; scaffolding is also used to support structures which might collapse
shovels and pickaxes, as well as trowels
water - for drinking
sample boxes (sturdy plastic or wooden crates)
surveying equipment - depending on the period of interest, a total station or a theodolite, and an auto-level or Dumpy level
grid markers - metal pins in the ground or stakes with string delimiting a grid or individual trenches
photography scales (1 m long banded in red and white, can be joined together for greater lengths)
cameras
piles of notebooks and record sheets
drawings
a hoist (only in particular cases, but it's a nice accessory to add to the mix)
a drone (that's my job
) - obviously not for earlier digs, although you could add a balloon for aerial photos!
a storage shed (we use a container in one particular site) which can be locked to keep tolls safe
wheelbarrows
buckets, for removing soil and other material removed
sieves
drawing boards
grids (for drawing small areas to scale)
the odd laptop
archaeologists(!), with sturdy boots, gloves, hats
ideally, a support vehicle (or more than one)
sandbags, for shoring loose soil or sediment; or for backfill (filling up the trenches if conservation is not to be carried out)
wasp's nests (at least we always get these)
fencing around the site - to keep people from mucking about, falling in pits or grave robbing!
Amongst the stuff generated by a dig would be:
trenches - not necessarily military style, shallow rectangular ones or sampling pits are possible
the site itself - many trenches reveal just bedrock, further layers of soil, or some sort of paving, which can be removed if necessary; you also get structures of course, mostly stone, unless conditions are good for preserving wood
a spoil heap - the material removed - in many cases, one for soil, one for stones
architectural features (stone) which are removed from where they are found to be conserved or transported, or simply stored near the site
finds - pottery being the commonest, mostly broken, some intact
visitors and the press, including film crews!
backfilled areas - trenches which have been refilled; in modern archaeology, geotextile is used, and some inert material which is different from the ground material is piled in, usually in sacks or sandbags; in less well-funded circumstances the material removed is placed back, although some sort of textile is placed as a boundary between excavated and replaced material
In sensitive areas which are of major relevance, a security guard may be on site - usually with some sort of shed or office.
More specialized locations might need taller scaffolding with platforms and planking, more permanent guard rails around deeper holes, caving or even diving equipment, portable analytical equipment etc.