We experimentally demonstrated Intensity-Modulated Direct-Detection (IMDD) single-channel 1,040 km linear transmission and 800 km non-linear transmission at 10 Gb/s over standard single-mode (G.652) fiber, without any optical dispersion compensation or mitigation, using a Maximum-Likelihood Sequence-Estimation (MLSE) receiver employing the square-root (SQRT) branch metric with off-line processing. These experiments were designed as to probe the limits of the MLSE approach. They successfully showed that long-haul uncompensated transmission is in principle possible with MLSE, even in the presence of large uncompensated dispersion and strong intra-channel fiber non-linearities, provided that enough complexity can be built into the receiver. In the linear 1,040 km experiment, a Bit Error Rate (BER) of 10(-3) was achieved with an Optical Signal-to-Noise Ratio (OSNR) penalty with respect to back-to-back of 2.9 dB, using two samples per bit and 16,384 trellis states. Several other set-ups were tested as well, including the use of only one sample per bit and fewer trellis states. In the non-linear 800 km experiment, power was ramped up to 12 dBm, exciting substantial Kerr non-linearity, whose induced spectral-broadening exacerbated the effects of the large uncompensated dispersion of the link. Using an MLSE receiver with 1,024 states, we demonstrated a non-linear threshold of 9 dBm. We benchmarked this experiment towards simulations addressing various electrical and optical dispersion compensation strategies. We also carried out an analysis of error run-lengths, on both experiments, which showed that error burstiness may change considerably depending on the number of processor states, OSNR and the amount of non-linearity in the link.