Intro
TimeBlur is one of my favourite nodes to play around with, just because it allows you to utilise sub-frame information. This means that expressions that operate between integer frame values can become incredibly powerful tools.
The most basic introduction to this is 'frame % 1'. This simply creates a sawtooth ramp from 0 to 1 between frames.
To invert this, do as you would in nuke, and use '1 - (frame % 1)'
So if we want to create an exponential curve using this principle, we could square our modulus expression (frame%1) to give an exponential curve.
Practical Applications
One of the most common uses of exponential-eque processes in compositing is an 'Exponential Blur'. Bloom is an example where exponential blur comes in handy. The maths behind light falloff is follows the Inverse Square Law .
By using an expression in the size value on a blur node and placing a timeblur beneath it, we can average out that expression over the frame to create a soft blurred result.
This exponential ramp up already gives a more convincing glow effect than a standard blur and ,thanks to TimeBlur's 'divisions' slider, more steps can be added without adding more nodes.
Demo
I've put together a quick demo set-up to demonstrate how combining multiple methods can lead to a convincing result, without needing to include multiple blur nodes. You can download it here
Disclaimer: As with all node based solutions, there's unlimited ways to solve the same problem. This is just one experiment to see what's possible. I'm certain there are simpler and more efficient solutions to exponential blur, but boy if we did have fun along the way!