The pdf lists an example of fitting a model with crossed random effects using the
Penicillin dataset in section 2.1.2 (p. 4 of pdf), and an example of fitting a model with nested random effects using the
Pastes dataset in section 2.2.2 (p. 14 of pdf). At the beginning of the latter section, it reads:
Fitting a model with simple, scalar random effects for nested factors is done in exactly the same way as fitting a model with random effects for crossed grouping factors.
I don't know much about your data, but from these I would guess that you might specify your model like so:
model = lmer(y ~ 1 + covariates + (1|Sample) + (1|typeStatus) + (1|day), data=myData)