Any context where event sourcing is a good fit is a good fit for Akka Persistence.
Event sourcing, in turn, is generally applicable (note that nearly any DB you're using is event sourcing (with exceptionally frequent snapshotting, truncation of the event log, and purging of old snapshots)).
Event sourcing works really well when you want to explicitly model how entities in your domain change over time: you're effectively defining an algebra of changes. The richer (i.e. the further from just create/update) this model of change is, the more it's a fit for event sourcing. This modeling of change in turn facilitates letting other components of a system only update their state when needed.
Akka Persistence, especially when used with cluster sharding, lets you handle commands/requests without having to read from a DB on every command/request (basically, you'll read from the DB when bringing back an already persisted actor, but subsequent commands/requests (until such time as the actor passivates or dies) don't require such reads). The model of parent and child actors in Akka also tends to lead to a natural encoding of many-to-one relationships.
In the example of a trello board, I would probably have
- each board be a persistent actor, which is parent to
- lists, which are persistent actors and are each parents to
- list items, which are also persistent actors
Depending on how much was under a list item, they might in turn have child persistent actors (for comments, etc.).
It's probably worth reading up on domain-driven design. While DDD doesn't require the actor model (nor vice versa), and neither of them requires event sourcing (nor vice versa), I and many others have found that they reinforce each other.