We study cluster sizes in supercritical $d$-dimensional inhomogeneous percolation models with long-range edges -- such as long-range percolation -- and/or heavy-tailed degree distributions -- such as geometric inhomogeneous random graphs and the age-dependent random connection model. Our focus is on large deviations of the size of the largest cluster in the graph restricted to a finite box as its volume tends to infinity. Compared to nearest-neighbor Bernoulli bond percolation on $\mathbb{Z}^d$, we show that long edges can increase the exponent of the polynomial speed of the lower tail from $(d-1)/d$ to any $ζ_\star\in\big((d-1)/d,1\big)$. We prove that this exponent $ζ_\star$ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin $\mathcal{C}(0)$. For the upper tail of large deviations, we prove that its speed is logarithmic for models with power-law degree distributions. We express the rate function via the generating function of $|\mathcal{C}(0)|$. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.