For no information to be lost on conversion back to continuous form, the signal would first need to be perfectly band-limited, and you would need an ideal reconstruction filter. A perfectly band-limited signal is infinite in extent. Since you want this arbitrary signal to be processed by a computer, your computer would need infinite memory. You would also have to wait an infinite amount of time for your perfect reconstruction filter to settle.
Finite sampling time and filter length problems also completely ignores other potential lossy issues, such as quantization and sampling clock jitter, etc.
One might be able to process an infinite signal that can be represented in certain closed symbolic forms within finite time and resources, but there does not seem to be a method to convert any arbitrary signal into such a form.
Back in the real-world (stuff you can buy or make), one would normally accept an information loss from imperfect band-limiting, finite filter length, jitter (etc.) that is around or below the quantization and numeric noise floors. This allows the processing to happen using an amount of RAM that one can afford to hopefully finish within one's lifetime. Thus leading to information loss and imperfect reconstruction.