In clinical trials, studies often present longitudinal data or clustered data. These studies are commonly analyzed using linear mixed models (LMMs), usually considering Gaussian assumptions for random effect and error terms. Recently, several proposals extended the restrictive assumptions from traditional LMM by more flexible ones that can accommodate skewness and heavy-tails and consequently are more robust to outliers. This work proposes a canonical fundamental skew-t linear mixed model (ST-LMM), that allows for asymmetric and heavy-tailed random effects and errors and includes several important cases as special cases, which are presented and considered for model selection. For this robust and flexible model, we present an efficient EM-type algorithm for parameter estimation via maximum likelihood, implemented in a closed form by exploring the hierarchical representation of the ST-LMM. In addition, the estimation of standard errors and random effects is discussed. The methodology is illustrated through an application to schizophrenia data and some simulation studies.