A person starts running at 1m/s
Every metre they run, they gain 1m/s in speed. What is a function that describes their speed at time t (metres per second) and their distance at time t (in metres)?
Assuming the person can run for as long as possible, laws of physics don't apply, and they don't have a speed cap.
Using @garondal's comment $$t_d=\sum_{i=1}^d \frac{1}{i}=H_d$$ there is an approximate inverse of the harmonic number (have a look here).
Using @Gary's comments, $$d \sim x-\frac 12 -\frac 1{24 x}+\frac 3{640 x^3}+O\left(\frac{1}{x^5}\right)\qquad \text{with} \qquad x=e^{t_d- \gamma }$$