Do there exist functions $t\to f(t)$ so that $f(0) = +\infty$ but decaying so fast that the Riemann integral $\int_0^1 f(t)dt$ converges? Maybe my memory betrays me, but I can't think of any such function from my calculus school days.
Wait a minute... isn't $f(t) = -\log(t)$ such a function?
$$\int -\log(t)dt = \left[t-t\log(t)\right]_0^1$$ Which converges by famous limit $$\underset{t\to 0}{\lim}\left\{t\log(t)\right\}=0$$
Do there exist many other examples expressible as (integrals of) elementary functions or is it a special case?
Yes, there are such examples. For example, let $f(x) = 1/\sqrt{x}$, then $$\int_0^1\frac{1}{\sqrt{x}}\,\mathrm d x = 2 $$
In fact, in general if you choose $g_p(x) = 1/x^p$, then the integral $$\int_0^1 g_p(x)\,\mathrm d x = \frac{1}{1-p} $$ and converges for all $0<p<1$.