Has anyone used FPGA chip in a project such as a simple calculator, that does not use the devKit/board only the chip plugged in your own circuit?
FPGAs are a poor choice for implementing applications like calculators which involve sequential computation but don't need the parallelism, deterministic timing, and speed of FPGAs. Calculators, or applications that need slow human interfaces and slow but complex calculation are best implemented using MCUs. Yes, you can implement an MCU within an FPGA or do all of the computation in parallel logic, but the result will be far more expensive, power inefficient, and complex than using an MCU.
Ideal applications for FPGAs are where you do need parallelism, deterministic timing, and speed for which MCUs are poor choices. These applications include interfaces, bus bridges, translation from one protocol to another at line speed, signal processing, precise timing, parallel computation, etc.
Past projects I've done included data acquisition boards where the timing of the front end muxes, gain, offset, A/D converter, FIFO were timed to achieve optimal performance with minimal switching noise, parsing Ethernet TCP/IP packets in parallel logic, custom DDR SDRAM interface with scatter-gather DMA, multi-channel galvonometer vectoring with laser timing and modulation, mimicking 6521 PIAs interfaced to a legacy 6502 based instrument with EPI interface to an ARM Cortex-M3 MCU, PCIe interface with DMA, peripheral implementing stepper motor control with encoder position feedback, proprietary parallel and serial bus interfaces with scatter-gather DMA, and so on. All of these applications would be very inefficient to implement with normal processors if even possible.
A good advanced FPGA project for students has always been to implement a VGA controller. A VGA controller requires maintaining accurate timing, DMA, state machines and also possibly configuration, control, and status. Faults and successful implementation are very observable. A very useful addition would be to implement it with an SPI interface, which would permit you to easily reuse it with MCUs. Many educational DevKits have the VGA connector with either onboard DAC or resistor network.