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I have a laptop with a 100mb/s ethernet and a 75mb/s wireless connection to my internet router. This is the maximum speed each network adapter can get in my laptop, but my external link is higher.

I'm thinking if it's possible to use both connections to my router as one, joining their capacity up to 175mb/s in theory. My router is a Linux box, like most of the wifi routers, and I have ssh access to it, thus allowing it possibilities for some setup there as also.

Here's a graph of the network setup:

 +----------+ETH (100Mbs)<------>-----------+Fiber (300Mbs)<-------> External
 | MY | | ROUTER |
 | LAPTOP | | |
 +----------+WIFI (75Mbs)<------>-----------+

I know that there are other ways like buying an external gigabit ethernet adapter. But this is not a huge problem. I was just wondering if this is possible so setup (with no additional cost), so I could have better use of what I have, and I found out it's an interesting question.

Simply: No, you cannot achieve that.

You may consider a solution that is easier: you have 2 ethernet interfaces and want to use them both. In Linux, it is called "bond" which uses LACP for bonding interfaces into one logical. However this requires support on the side of the switch as well (it needs to be configured, not sure if home switches are capable of such a thing). Even then one single connection will not be able to pass through more than a single link capacity (it is based on hashes of some header fields - which in communication between 2 nodes and the same service will be the same).

Another problem is routing because you have two separate L3 ports, your routing computation needs to be deterministic - therefore you can choose only one interface at a time for your usage.

What you could achieve is: consider you are running a few virtual machines, you will grant them access to ethernet (bridge the interface) and you will continue to use the wifi only. This way your throughput could rise over the limit of ethernet, but is a very specific scenario.