At the end of the day, the digital circuitry in an FPGA chip are analog circuits. When they are operating within their bounds, they properly behave like ideal digital circuits. Usually you want your digital parts to operate digitally but some of the analog characteristics might be useful, according to Alex Lao on the Voltage|Divide blog. The items Alex looks at:

• Process, Voltage and Temperature Variation
• Propagation Delay
• Combinational Logic Glitching
• Metastability
• Overclocking
• Device Primitives
• Ring Oscillators
• Dynamic Propagation Delay Measurement
• Strain Gauge
• Time to Digital Converters
• Tapped Delay Lines
• Short Pulse Generator
• True Random Number Generation
• Physical Unclonable Functions
• Digital to Analog Conversion
• Analog to Digital Conversion
• Frequency Synthesis
• Antenna

See the complete discussion in the post here.