Mixed effects model for longitudinal data with nested random effects? I have three groups, one control, a low dose treatment, and a high dose treatment.
These groups are composed of 6 subjects nested in each group, for a total of 18 subjects.
Each of these 18 subjects is measured for 7 days at 1 day intervals.
My goal is to find out if there is significant difference in the outcome variable between the three treatment groups. I could use a mixed effects model to do this, but how? Would it be something like:
lmer(outcome ~ day + (1|group) + (1|subject)?

Assuming that each subject is only in one group, you have a nested design. Conceptually, it makes more sense to treat group as a fixed effect. As you have only three groups, it wouldn't make much sense statistically speaking anyways because you'd be asking the software to estimate a variance for group assuming a normal distribution from only three observations.
Your sample size is quite low which might be a problem. Nevertheless, I suggest starting with a simple random intercept model:
lmer(outcome~day + group + (1|subject), data = dat)

This model fits a global intercept which is simply the intercept for the reference group, deviations from that intercept for the remaining groups, a single slope for the effect of day and a random intercept for subject. Hence, this model assumes that each group has the same trajectory over time, the same slope but different intercepts. To allow each group their own slope, you could fit the following model with an interaction between day and group:
lmer(outcome~day*group + (1|subject), data = dat)

Lastly, you could also allow for random slopes:
lmer(outcome~day*group + (day|subject), data = dat)

All these models assume a linear relationship between the outcome and day which might be unrealistic. If you suspect nonlinear relationships, you could easily accomodate those by using polynomials or (my recommendation) restricted cubic splines (aka natural splines) with, say, 3 knots. Finally, have a look at this post which goes into more details about such longitudinal models.
