It varies heavily, but this term it looks as if I will have between ten-twelve essays, one presentation, one research design, and one field report. In addition, I have my dissertation to research (thus why I'm at the British Museum on Wednesdays). There are, in addition, two assessed essays which are due at around the same time as the dissertation.
I wouldn't know what constitutes a lab report, as such. For excavation there is a formal system of recording and reporting which is used to convey data into a published format, which are largely 'grey literature' and heavily jargon-based*. This style of report is normally produced by trench supervisors/excavation directions, derived from context sheets which are produced by the excavators in the trenches (i.e., me). I produce one field report this year, which is pretty much like the quotation below. Otherwise, essays tend to be synthetic, and, as such, do not need to enter into the same degree of descriptive prose that a report would feature. Sure, you might occasionally have to perform close-studies of stratigraphic reports, like that below, but it depends on your specialisms: I'm a ceramicist, so I have my own world of jargon that would probably be incomprehensible to a lithics specialist (it's all "vegetable- and chaff- tempered micaceous wares developed by SSC", and so on).
Oh, and of course, we too have densely ontological theoretical essays.
* to the tune of: "Locus 1875 consisted of five superpositioned-stratigraphic layers, beginning with Context t. 5 which is a 10 cm lens of ashy spoil containing larger grains of unidentified carbonized material. This layer occurred evenly throughout Locus 1875, and lay on a sterile basal level of sand-washed gravel and compacted, yellowish fill ..." etc, etc, etc.