Sorry for the stupid title. When we define ind-objects in a category, we use in general filtered diagrams in a category, not just sequences $A_1 \rightarrow A_2 \rightarrow A_3 \dots$ indexed by the integers. Why?

I'm not doubting there is a good reason. What I would really like is some "weird" examples of filtered categories to keep in my mind, which behave very differently to the positive integers as a directed set.

I edited the question in view of several helpful replies (thanks).

When we define ind-objects in a category, we use in general filtered diagrams in a category, not just sequences $A_1 \rightarrow A_2 \rightarrow A_3 \dots$ indexed by the integers. More generally, if one wants a "bigger" limit, we could look at diagrams indexed by an ordinal.

Is there a simple example of an ind-object that can't be indexed by an ordinal?