This paper studies a single-cell uplink massive device communication scenario in which a large number of single-antenna devices are connected to the base station (BS), but user traffic is sporadic so that at a given coherence interval, only a subset of users are active. For such a system, active user detection and channel estimation are key issues. To accommodate many simultaneously active users, this paper studies an asymptotic regime where the BS is equipped with a large number of antennas. A grant-free two-phase access scheme is adopted where user activity detection and channel estimation are performed in the first phase, and data is transmitted in the second phase. Our main contributions are as follows. First, this paper shows that despite the non-orthogonality of pilot sequences (which is necessary for accommodating a large number of potential devices), in the asymptotic massive multiple-input multiple-output (MIMO) regime, both the missed detection and false alarm probabilities can be made to go to zero by utilizing compressed sensing techniques that exploit sparsity in user activities. Further, this paper shows that despite the guaranteed success in user activity detection, the non-orthogonality of pilot sequences nevertheless can cause significantly larger channel estimation error as compared to the conventional massive MIMO system, thus lowering the overall achievable transmission rate. This paper quantifies the cost due to device detection and channel estimation and illustrates its effect on the optimal pilot length for massive device connectivity.