Does hydrogen really have seven isotopes?
Yes, there are seven known isotopes of hydrogen, though only two ($\ce{^1H}$ and $\ce{^2H}$) are stable with respect to nuclear decay, and only three ($\ce{^1H}$, $\ce{^2H}$ and $\ce{^3H}$) exist/can be made in enough quantities to be relevant outside of nuclear physics. All other hydrogen isotopes have extremely small half-lives. The next most stable hydrogen isotope after $\ce{^3H}$ ($t_{1/2}=12.3\ yr$) seems to be $\ce{^5H}$ ($t_{1/2} \sim 10^{-21}\ s$). Given that even the fastest chemical reactions happen in timespans of the order of $10^{-15}\ s$, we can safely discard any influence of hydrogen isotopes heavier than $\ce{^3H}$ in chemistry.

Out of the currently 115 elements which have been confirmed to exist, there are approximately 254 isotopes which are considered to be indefinitely stable, or have half-lives so enormous (over trillions of times greater than the age of the Universe, 13.8 billion years) that no value could yet be determined. Theoretically, only 90 of those 254 isotopes are expected to be truly indefinitely stable (proton decay aside), but there are great difficulties in setting up experiments capable of measuring such rare decay events. 

On top of the previous 254, another 34 isotopes have half-lives which have actually been measured in laboratory, but are still large enough (>50 million years) so that a small fraction of the isotope could conceivably have survived since the creation of the Earth 4.6 billion years ago, and may be of interest in geological timescales. There are also isotopes with smaller half-lives, but which are continuously replenished by the decay of heavier atoms or by the action of cosmic rays, so they can still be found on Earth today. In all, about 339 different isotopes can be found naturally on our planet.

However, as many as 3100-3300 different isotopes of the first 118 elements are claimed to have been detected in laboratories, most of them with very small half-lives (seconds or less). Nobody knows for sure how many elements and how many isotopes can exist, though the number is finite.