One other time when aliasing isn't a problem is when designing lowpass filters used for decimation. You can allow some amount of aliasing after the decimation operation to relax the constraints on the filter's performance, resulting in a lower-order design. Instead of placing the stopband edge at the post-decimation Nyquist frequency, you can slide it out just far enough that it doesn't alias back into the filter's passband (and therefore corrupt your signal of interest).

Stated more mathematically, assume that your original signal is sampled at rate $f_s$ and you are decimating by a factor $D$. The filter's passband edge $f_p$ is defined by the bandwidth of the signal of interest. The Nyquist frequency after decimation will be $\frac{f_s}{2 D}$, so obviously the passband edge frequency must be less than this.

Since I've asserted that you can allow the stopband to bleed past the decimated Nyquist frequency, recall how aliasing in the second Nyquist zone works: any content at frequency $\frac{f_s}{2 D} + \Delta f$ before the decimation operation will appear to be located at $\frac{f_s}{2 D} - \Delta f$ afterward. Thus, we can place the stopband edge at some frequency $f_{stop} = \frac{f_s}{2 D} + \Delta f$ and select $\Delta f$ in such a way that the filter's transition band doesn't overlap with the passband after we decimate. In order for this to be true:

$$ f_{stop_{aliased}} = \frac{f_s}{2 D} - \Delta f \ge f_p $$

$$ \Delta f \le \frac{f_s}{2 D} - f_p $$

The takeaway from this is if there is still a decent amount of oversampling present in the post-decimated signal (there are some reasons why you would do this), then you can push the stopband out by a nontrivial amount. As a quantitative measure, you can look at the transition ratios of the "naive" and "relaxed" filter specifications:

$$ T_{naive} = \frac{f_p}{f_{stop_{naive}}} = \frac{f_p}{\frac{f_s}{2 D}} = \frac{2 D f_p}{f_s} $$

$$ T_{relaxed} = \frac{f_p}{f_{stop_{relaxed}}} = \frac{f_p}{\frac{f_s}{2 D} + (\frac{f_s}{2 D} - f_p)} = \frac{f_p}{\frac{f_s}{D} - f_p} $$

$$ \frac{T_{relaxed}}{T_{naive}} = \frac{\frac{f_p}{\frac{f_s}{D} - f_p}}{\frac{2 D f_p}{f_s}} $$

$$ \frac{T_{relaxed}}{T_{naive}} = \frac{1}{2 - \frac{f_p}{\frac{f_s}{2 D}}} $$

This last expression gives you a compact representation of the improvement in transition ratio that can be obtained by relaxing the filter specification in this way, parameterized by the ratio of the filter's passband (i.e. the signal of interest's bandwidth) to the post-decimation Nyquist frequency. Plotting this ratio as a function of the passband frequency (normalized by the post-decimation sample rate), you get:

So in summary, if your signal is still decently oversampled after the decimation operation, then you can reduce the filter's transition ratio by a factor of up to 2 by relaxing its specification in this way. As a rule of thumb, the number of required taps for an FIR filter is roughly proportional to the transition ratio. It does allow some aliasing when performing the decimation, but the specifications are designed such that the aliasing does not overlap with the desired signal. This allows it to be removed later if needed, by a filter that operates at the decimated sample rate $\frac{f_s}{D}$.

One other time when aliasing isn't a problem is when designing lowpass filters used for decimation. You can allow some amount of aliasing after the decimation operation to relax the constraints on the filter's performance, resulting in a lower-order design. Instead of placing the stopband edge at the post-decimation Nyquist frequency, you can slide it out just far enough that it doesn't alias back into the filter's passband (and therefore corrupt your signal of interest).

Stated more mathematically, assume that your original signal is sampled at rate $f_s$ and you are decimating by a factor $D$. The filter's passband edge $f_p$ is defined by the bandwidth of the signal of interest. The Nyquist frequency after decimation will be $\frac{f_s}{2 D}$, so obviously the passband edge frequency must be less than this.

Since I've asserted that you can allow the stopband to bleed past the decimated Nyquist frequency, recall how aliasing in the second Nyquist zone works: any content at frequency $\frac{f_s}{2 D} + \Delta f$ before the decimation operation will appear to be located at $\frac{f_s}{2 D} - \Delta f$ afterward. Thus, we can place the stopband edge at some frequency $f_{stop} = \frac{f_s}{2 D} + \Delta f$ and select $\Delta f$ in such a way that the filter's transition band doesn't overlap with the passband after we decimate. In order for this to be true:

$$ f_{stop_{aliased}} = \frac{f_s}{2 D} - \Delta f \ge f_p $$

$$ \Delta f \le \frac{f_s}{2 D} - f_p $$

The takeaway from this is if there is still a decent amount of oversampling present in the post-decimated signal (there are some reasons why you would do this), then you can push the stopband out by a nontrivial amount. As a quantitative measure, you can look at the transition ratios of the "naive" and "relaxed" filter specifications:

$$ T_{naive} = \frac{f_p}{f_{stop_{naive}}} = \frac{f_p}{\frac{f_s}{2 D}} = \frac{2 D f_p}{f_s} $$

$$ T_{relaxed} = \frac{f_p}{f_{stop_{relaxed}}} = \frac{f_p}{\frac{f_s}{2 D} + (\frac{f_s}{2 D} - f_p)} = \frac{f_p}{\frac{f_s}{D} - f_p} $$

$$ \frac{T_{relaxed}}{T_{naive}} = \frac{\frac{f_p}{\frac{f_s}{D} - f_p}}{\frac{2 D f_p}{f_s}} $$

$$ \frac{T_{relaxed}}{T_{naive}} = \frac{1}{2 - \frac{f_p}{\frac{f_s}{2 D}}} $$

This last expression gives you a compact representation of the improvement in transition ratio that can be obtained by relaxing the filter specification in this way, parameterized by the ratio of the filter's passband (i.e. the signal of interest's bandwidth) to the post-decimation Nyquist frequency. Plotting this ratio as a function of the passband frequency (normalized by the post-decimation sample rate), you get:

So in summary, if your signal is still decently oversampled after the decimation operation, then you can reduce the filter's transition ratio by a factor of up to 2 by relaxing its specification in this way. As a rule of thumb, the number of required taps for an FIR filter is roughly proportional to the transition ratio. It does allow some aliasing when performing the decimation, but the specifications are designed such that the aliasing does not overlap with the desired signal. This allows it to be removed later if needed, by a filter that operates at the decimated sample rate $\frac{f_s}{D}$.