Absolute minimum number of data points required for estimating random slopes for continuous covariates What is the absolute minimum number of data points needed to estimate random intercepts and random slopes for a continuous covariate? 
IIUC, in the 'categorical case' one needs at least two data points per subject to estimate random intercepts and at least two observations for each combination of subject and treatment level to estimate random slopes.
Does the same apply to continuous covariates, so that one would need two data points per subject to estimate random intercepts and one more, i. e. three, to estimate random slopes? 
 A: For answering this question, I feel it is important to remind that regression-type models, such as mixed models, really do not treat categorical covariates and continuous covariates any different on the most basic level (i.e., when estimating the model). Any such difference is only 'interpretational sugar' added by the software to make the analysis easier to understand.
What lme4 does for categorical covariates is to transform them into numerical covariates (using model.matrix()) and then use those numerical covariates in the optimization process. Because categorical covariates are transformed into numerical covariates (usually with two non-zero levels only) for estimation, what holds for those also needs to holds for continuous covariates. Therefore, you need exactly the same number of trials to make a specific random-slope identifiable whether it is continuous or categorical (as long as the categorical has two level only).
However, there is a further consideration to take into account. How many data points do we usually need to estimate a continuous covariate? If two points are far enough apart on the x-axis and estimated with enough precision, two data points may be enough. However, in most cases neither might be the case. What a random slope for a continuous covariates essentially is, is a unique slope estimate per ID (assuming hierarchical shrinkage). Estimating individual-level regression slopes with only two data points seems somewhat questionable. So I encourage you to think about this issue when determinign whether or not to estimate random-slopes. Can we with this amount of data estimate individual-level effects?

Update: I realize I have previously given out potentially false information. To estimate a full random-effects structure, you need one data point more than cells per individual. Two for random-intercepts and three for random-intercepts plus random slopes.  See following example which also shows no difference between continuous and categorical covariate:
df <- expand.grid(id = 1:20, rep = 1:4)

df$x <- rnorm(nrow(df))
df$X <- factor(df$rep %% 2)
df$y <- rnorm(nrow(df))

library("lme4")

m1 <- lmer(y ~ x + (x|id), df) # works
m1b <- lmer(y ~ X + (X|id), df) # works

df2 <- subset(df, rep < 4)
m2 <- lmer(y ~ x + (x|id), df2) # works
m2 <- lmer(y ~ X + (X|id), df2) # works

df3 <- subset(df, rep < 3)
m2 <- lmer(y ~ x + (x|id), df3) # fails
m2 <- lmer(y ~ X + (X|id), df3) # fails

