There are soo many projects, field of applications, platforms, architectures and tools to mention about but let me put here a few.
1- Begin with Audio projects, as probably every DSPeng. did so.
There's an open source program AudaCity which involves very rich set of audio functions implemented. So write your own, add your own, develop your own.
Again, audio dsp effects is an area rich for new and better algorithms. Try developing your own.
Digital audio synthesizers is another rich field with immense set of applications.
2- Do image video processing applications. If you are into data compression as well I highly suggest you FFMPEG project. Get into such a project and develop your own applications. There are many filters, effects, algorithms etc that are implemented and you can definietely add your own.
All these require nothing but a PC to develop and implement.
3- In addition, you can search hardware based projects too but they will require a more elaborate set-up. Try implementing some embedded systems using development boards. ADI, ST, TI, NI, NXP provide very convenient, versatile application development boards using their own microcontrollers, microprocessors and DSP procesors, multimedia processors and sensors.
4- Look for some SDR (software defined radio) for most affordable digital communications applications. RTL-SDR is its home. Get a usb-sdr dongle (a hack of DVB-T receiver) and write your own pure C-code decoding algorithm there.
5- A raspberry pi 3 with a pi-camera can provide you a real opportunity to deal with raw camera processing. You can develop a lot of video coding algorithms there.