It's not necessarily tiny. You could use an electrolytic capacitor with a capacity of a few millifarads, for example. However, the tolerance of larger capacito

It's not necessarily tiny. You could use an electrolytic capacitor with a capacity of a few millifarads, for example. However, the tolerance of larger capacitors is larger (usually 20%, and up to 50%) so that quickly becomes expensive. It's more likely that you'll generate a 10 second or less pulse with the 555 and then make it longer using (for example) T flip-flops. A single 744040 ripple counter IC for example can divide the output by up to 4096.