Here are some 'recipes' for mixing darks without using black. Many artists recommend "mixing your darks" rather than using black. It can help to keep your painting harmonious to mix your darks using pigments that are used elsewhere in your painting.
1. ultramarine blue and burnt sienna
2. alizarin crimson and viridian green
3. dioxazine purple and sap green
4. neutral tint and another colors
Things to think about are:
1. Use at least one color that is relatively dark to begin with. In the examples above, ultramarine blue, alizarin crimson, and dioxazine purple are all quite dark. The colors mixed with them are also moderately dark: burnt sienna, viridian, and sap green.
2. Another tactic is to use 'neutral tint' to deepen another color. Neutral tint isn't exactly neutral - it's quite bluish/on the cool side. But it can be an easy way to get some dark values.