My primary reason for raid is expansion.
I prefer not to have a whole bunch of drives connected to my machine
I assume you mean that you want to connect just one enclosure, which may have multiple drives.
A single disk enclosure which allows you to change disks might meet your needs. if you have 4 TB of data, you could purchase a 10 TB disk which would give you more than 2x as much storage capacity as you have now. This could be the cheapest and easiest way to go, although you would need a second enclosure (ideally from a different vendor and with a different vendor's drive) for your backups and for repopulating the main disk if you upgrade to a higher capacity drive. You would, of course, be limited to the read/write rates of the single drive.
In many hardware based RAID solutions an upgrade is not a simple thing. When you purchase a populated RAID it will likely have identical disks with the same capacity. If you have 4 TB disks and add a 6 TB disk to a RAID 5 the extra 2 TB won't be added to the RAID volume.
The Drobo series of disks is software based, so it does allow you to add disks of different capacities which can be used, but again there are limitations. See
http://www.drobo.com/storage-products/capacity-calculator/
If you populate with 4 TB drives (14.52 TB in the RAID 5 volume) and then add a 6 TB drive there will be no increase in capacity. You just get more "reserved for expansion", whatever that means. If you add a second 6 TB drive you get full use of the extra capacity of both drives, giving you 16.34 TB for 2x6TB and 3x4TB.
The process for upgrading a hardware RAID 5 is very painful. I just upgraded the drives in my primary RAID 5 device from 2 to 6 TB. The first pain was the purchase cost of buying 6 x 6 TB drives. Then you have to back everything up. You replace the drives in the unit. This can be as simple as ejecting and inserting the drives, or you many have to remove the old drive from a carrier and then put the larger drive in the carrier. Then you have to format the drive for RAID. This took almost 24 hours. Then you have to restore from your backup which could take another day.
One of the drives that I purchased failed and it took 2 weeks to get it replaced. Needing to use the drive I had to format with a lower capacity, and will have to do everything all over now that the replacement drive has arrived. And I stripped a screw in the carrier and can't remove the old drive from the carrier. So I have to get that fixed so I can use the replacement drive.
I'm not that familiar with software raid and other vendors, so maybe others can comment.
One solution you might want to consider is a RAID capable device but just set it up as a JBOD. You could start out with just two large JBOD disks (1 primary, 1 backup). If you need more capacity than a single disk you could use other slots for 2 RAID 0 volumes (1 for backup). Note, however, that there are risks in using RAID in one enclosure. If the hardware fails you can't just take the disks out and put them into another vendor's device since RAID formatting is normally specific to the hardware. If you have to purchase a new unit from the same vendor there is a risk that it might not recognize the disks if the hardware (and thus disk formatting) has changed. So it's probably best to have 2 drive enclosures once you move beyond JBOD to RAID.