Ours allows you to invest as much as you want, subject to the fact that you can only invest after you have $5000 in the account. The plan we had before this one set the limit at $6000. It varies by plan, but AFAIK, you can invest as much as you want, subject to any minimum cash balance requirement.
The way we work it is this: I keep our yearly OOP max in cash in case of catastrophe. Invest the rest. But we pay most of our regular medical bills out of out checking, and just keep the HSA growing. If we had to touch it we would, but since our medical expenses are minimal, we prefer to just let that money grow untouched until we really need it, or we reach retirement age and can then use it for anything. Basically, we treat it as an extra IRA or 401k.