This paper studies the performance of a transmission and reception scheme for massive access under some practical challenges. One challenge is the near-far problem, i.e., an access point often receives signals from different transmitting devices at vastly different signal strengths. Another challenge is that the signals from different devices may be subject to arbitrary, analog, and heterogeneous delays. This paper considers a fully asynchronous model which is more realistic than the frame or symbol level synchrony needed in most existing work. A main theorem characterizes the asymptotic scaling of the codelength with the number of devices, a device delay upper bound, and the dynamic range of received signal strengths across devices. The scaling result suggests potential advantages of grouping devices with similar received signal strengths and letting the groups use time sharing. The performance of the proposed scheme is evaluated using simulations with and without grouping.