If I HAD HAD more money...Can I say HAD HAD?

Yes, you can.  In the grammar world it makes up part of what is called the third conditional.  What's that?  Well, it's a way of expressing something that didn't happen, in the past, of course.  It's what we call...

WOULD HAVE 
COULD HAVE 
MIGHT HAVE

Or in more informal speech...woulda, coulda...mighta...

So where does the HAD HAD come in?  The HAD HAD goes with the IF clause.  Let's take a look at the structure first:

If + SUBJECT + HAD (NOT)+ Past Participle, ...SUBJECT + WOULD/COULD/SHOULD + (NOT) HAVE + Past Participle...

If I HAD driven slower I WOULD NOT HAVE gotten in the accident.

Here is an example of each using HAD HAD in the IF clause.  The first HAD is the conditional past perfect, and the second HAD is the past participle.

1.  If I HAD had more money, I WOULD HAVE bought a faster car.
2.  If I HAD had more time, I COULD HAVE spent more time at the beach.
3.  If I HAD had more experience, I MIGHT HAVE gotten that job.

Just so you know, us native speakers usually don't say HAD HAD.  Instead we use the following contractions to make it flow better:

I had had= I'd had
You had had= You'd had
He had had= He'd had
She had had= She'd had
It had had= It'd had
We had had= We'd had
They had had= They'd had

Well, that's it for now.  I hope this clarifies your doubts about English's idiosyncratic HAD HAD.