I'm spreadsheeting my favorite albums and I'd like the good tracks on the album to count more for its average than the bad songs. I thought about geometric means, but those do the opposite of what I want.
Is my best course of action to raise each value to a certain number and then sum them and divide by some constant?
The generalized $p$-mean will do roughly what you want:
$$A_p = \left(\frac{1}{N}\sum_{i=1}^N x_i^p\right)^{1/p}.$$
When $p=1$, you get the usual mean. The large $p$ you choose, the more the outliers will dominate the average (so that as $p\to\infty$, the album average approaches the maximum of the song scores.)