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This paper presents a non-orthogonal multiplexing scheme using the concept of rateless coding for a downlink broadcast scheme with two users. The proposed approach does not require the inclusion of successive interference cancellation (SIC) techniques for signal detection when compared to the power-domain non-orthogonal multiple access (PD-NOMA) schemes. In various user pair scenarios, it is shown that the proposed scheme not only enhances the user transmission rate but also reduces the overhead required for reliable transmissions when compared to PD-NOMA-based schemes. To further enhance the error performance when considering both unconstrained and constrained inputs, a code design algorithm is proposed based on the transmission rate analysis. It is shown that the designed rateless multiplexing schemes achieve error performances that accurately approach the predicted performance limits.