An Intel service employee once told me, that it is possible to add an additional HDMI port to my NUC. Unfortunately I can't find anything concerning this matter. Maybe somebody here could help me out.
This another option you can try: https://www.gorite.com/intel-nuc-dual-hdmi-lid-for-7th-generation-units
Your only option for getting a high-performance monitor interface is to use an adapter to convert the DisplayPort stream that is contained within the USB-C/TBT3 interface into an HDMI stream. There are all sorts of adapters that can do this. Choose carefully if you (also) want the ability to support a 4K monitor at 60Hz. Here's an example: https://www.amazon.com/Adapter-uni-Thunderbolt-Compatible-MacBook/dp/B07BT11PTP
A secondary option is to use an adapter that contains an MST Hub to convert that same DisplayPort stream into separate signals for multiple monitors. You can find such adapters that allow you to connect two or even three monitors and do so using either DisplayPort or HDMI interfaces. Al provided a link for one that offers support for connecting two HDMI monitors. Let's talk about the negatives. First of all, since the monitors will be sharing the bandwidth of the original DisplayPort stream, you certainly won't be able to support multiple 4K monitors at 60Hz. You can support two 4K monitors at 30Hz, however. You can also support multiple 1080p, 2K or even Ultra-Wide monitors at 60Hz, however. Secondly, Intel HD Graphics supports a maximum of three monitors. If you are using a MST Hub adapter with three outputs, you wouldn't be able to use the other HDMI port the NUC provides or, if you did, you wouldn't be able to support three monitors off the MST Hub adapter.
If you need to get beyond the three monitor limit of Intel HD Graphics, monitor interface(s) can be achieved via USB. There are USB-based graphics adapters that can provide one or two DisplayPort or HDMI output streams. The negative is that these streams are lower performance, consume CPU resources (a concern with only a Core i3) and most-definitely are not suitable for gaming. If all you want is a interface that can host your web browser, email or other office applications, these are just fine. Another issue is cost. These adapters are more pricy that they would be otherwise because they contain an actual Graphics controller. Leon has pointed you to a replacement lid for the NUC that provides two HDMI interfaces. This level of integration is great, but there is a problem. The included graphics controller is connected to the NUC's internal USB 2.0 ports. This is even lower performance that you could get if connected to a USB 3.0 port. The alternative is to use an external adapter(s) that plugs into one of the USB 3.0 ports on the back of the NUC. Here are some links to such adapters (I am using these simply as examples, this is not an endorsement):
- Single HDMI port: https://www.amazon.com/StarTech-com-External-Monitor-Graphics-Adapter/dp/B00OD37KHG
- Dual HDMI ports: https://www.amazon.com/StarTech-com-External-Monitor-Graphics-Adapter/dp/B0725K1MHH?th=1
Hope this helps,