Hi Chase, and welcome!
You have posted this in the Home Recording Forum, so when you say "mobile digital recording set-up" are you talking about something built into a vehicle or simply a wheeled rack of gear that can be lifted and placed in a van for transport to a recording location?
My "mobile" rig consists of a pair of Alesis HD24XR hard disc recorders coupled with a variety of pre-amps and montoring mixers, but that sort of system does not come in anywhere near $3K. You can indeed get 16 channels of pre-amps for $3K that will interface to a laptop, but putting that in the same sentence as the word "quality" is more difficult.
Does your "sufficiently powered" laptop have a FireWire port or an ExpressCard or CardBus slot? You can get 8-channel FireWire interface boxes that will daisy-chain to give you the 16 channels you mention, or you could get an 8-channel master interface unit that has expansion ADAT ports into which you could plug 8-channel pre-amp boxes with ADAT output.
Before comitting to purchase, one thing you ought to consider is whether you really want all 16 pre-amp channels to have the same sonic flavour. You haven't mentioned your microphones or the type of music you aim to record, but the availability of at least two different types of pre-amp combined with a good selection of different microphones adds greatly to your flexibility of what you can record and how it will sound. You are not going to get a huge variety with only a $3K budget, but you could consider 8 channels of transformer-input pre-amps and 8 with conventional electronic inputs.
Before making specific pre-amp equipment suggestions, it would help if you could tell us a little more about what you expect to record, what you have in your microphone closet and the type of laptop you intend to use.