Why can't I get a output lower than 0V in this diode and capacitor circuit?
Here's a step by step list of what happens:


The input starts positive. The diode is reverse biased, so no current flows and the capacitor doesn't charge.
Around \$t = 0.5s\$, the input becomes negative. The diode is forward biased, so the capacitor charges to \$1 V\$ (the negative of the minimum voltage applied). Note that there is no resistor, so there is no time constant - the capactor acts like an open circuit (as if the source is DC).
After the capacitor finishes charging, there is an offset of \$+1 V\$, and the diode never becomes forward biased again. The output is \$V_{in} + 1V\$.