Can I store information even if the trigger throws an exception?

As you've noticed if an apex script throws an unhandled exception everything it did is rolled back: @future, emails, DML, etc.

The good news is that there is a way to prevent a DML operation from succeeding without causing the whole apex transaction to be rolled back (except in the specific case mentioned below): the sObject.addError method. If you call this method on every sObject in your trigger it will prevent them all from being committed to the database, however any other DML operations you perform (that of course, do not cause errors in their own triggers) will be committed successfully.

If you're looking to log to a custom object this is your best bet; catch the exception, mark everything in Trigger.new with addError, then commit your log sObject.

EXCEPTION - To summarize the edits from subsequent contact with salesforce: this will not work if every record in the trigger has errors. In this case all work done in the trigger, including @future method calls, sending email, queueing batch jobs, or performing any DML, is rolled back.

I opened a Salesforce case with partner premier support (who actually troubleshoot apex issues!) #08522717. Salesforce claims this is working as designed, and pointed me to the wiki. Which seems to suggest that setting the all or nothing flag will impact this behavior but in my testing it didn't matter - if no DML rows commit without errors all operations are rolled back.

There is a very good wiki page on Developerforce.com on Exception handling. There you can see how to use email notifications on errors or even write an error record to a custom object.

http://wiki.developerforce.com/page/An_Introduction_to_Exception_Handling

Summer '17 now has Platform Events.

One of the interesting details about Platform Events from the docs:

Platform Events and Transactions
Unlike custom objects, platform events aren’t processed within database transactions in the Force.com platform. As a result, publishing platform events can’t be rolled back. Note the following:

• The allOrNoneHeader API header is ignored when publishing platform events through the API.
• The Apex setSavepoint() and rollback() Database methods aren’t supported with platform events.

Note the "can't be rolled back" part. Even if your top level transaction rolls back due to an exception you can still publish an event.

The pattern would look something like:

trigger ProblemTrigger on Account (before insert) {
    try {
        // ... trigger.new ... throws Exception
    } catch (Exception ex) {
        List<DebugDetails__e> errorMessages = new List<DebugDetails__e>();
        errorMessages.add(new DebugDetails__e(message__c = ex.getMessage()));
        List<Database.SaveResult> results = EventBus.publish(errorMessages);
        // Might need to handle railed publishing results
        throw ex; // Rethrow syntax?
    }

}


trigger CaptureDebugTrigger on DebugDetails__e (after insert) {    


    // Iterate through each notification.
    for (DebugDetails__e event : Trigger.New) {
        System.debug('Debug: ' + event.message__c);
        // Queue insert of the persistent object with the message details.
    }        
    // Insert the collection of error objects.
}

So you could publish and event with the required information and have a corresponding Apex trigger subscribed to the event to store the persistent message.