Is there a low-cost alternative to RJ45 cables for large number of signals?

There are ribbon cables with twisted wires:

Every few cm, there is a flat section to allow attaching an IDC connector.

Looking at two product drawings it appears to be pretty typical to have 50 mm of flat cable out of every 500 mm.

SCSI cables come to my mind.. they have lots of pins (68 in the ultrawide version), are well shielded, and can be gotten rather cheaply since nobody uses SCSI storage anymore. If you go for the LVDS type, the connectors are even acceptably small (although painful to route then).

You can use some form of ribbon cables. A cheap option is to use IDE cables, like those used by older PATA hard drives:

Each of those has 40 pins, so 5 of them gets you the 200 pins you want.

Another option would be to use direct pins to connect the boards. However, if the PC/104 standard is already a bit hard to connect (due to sum friction from all the 104 pins), then 200 will surely be harder to connect.

If you have more freedom when connecting the boards, you can also use a SODIMM (or even the older DIMM) slot and connect boards directly:

Some versions have 200 pins. By the way, this is exactly the same interface that the Raspberry Pi Compute Module uses (also 200 pins):

On the other hand, if you can design it in, it might be better to just serialize your data before getting it out of the board, unless your whole circuit is analog.