How many database indexes is too many?

It depends on the operations that occur on the table.

If there's lots of SELECTs and very few changes, index all you like.... these will (potentially) speed the SELECT statements up.

If the table is heavily hit by UPDATEs, INSERTs + DELETEs ... these will be very slow with lots of indexes since they all need to be modified each time one of these operations takes place

Having said that, you can clearly add a lot of pointless indexes to a table that won't do anything. Adding B-Tree indexes to a column with 2 distinct values will be pointless since it doesn't add anything in terms of looking the data up. The more unique the values in a column, the more it will benefit from an index.

I usually proceed like this.

1. Get a log of the real queries run on the data on a typical day.
2. Add indexes so the most important queries hit the indexes in their execution plan.
3. Try to avoid indexing fields that have a lot of updates or inserts
4. After a few indexes, get a new log and repeat.

As with all any optimization, I stop when the requested performance is reached (this obviously implies that point 0. would be getting specific performance requirements).

Everyone else has been giving you great advice. I have an added suggestion for you as you move forward. At some point you have to make a decision as to your best indexing strategy. In the end though, the best PLANNED indexing strategy can still end up creating indexes that don't end up getting used. One strategy that lets you find indexes that aren't used is to monitor index usage. You do this as follows:-

alter index my_index_name monitoring usage;

You can then monitor whether the index is used or not from that point forward by querying v$object_usage. Information on this can be found in the Oracle® Database Administrator's Guide.

Just remember that if you have a warehousing strategy of dropping indexes before updating a table, then recreating them, you will have to set the index up for monitoring again, and you'll lose any monitoring history for that index.