High pass filter of 0.01 Hz for EEG slow drifts
You're asking for a time machine: in order to avoid the DC bump at $t=0$, you need to know the state of the filter as if it had been running before the recording ever began. That's not going to happen no matter what kind of filter you use.

Here's a trick which may come in handy for you. Let's say you have an $N$-tap FIR filter (or an IIR filter whose impulse response decays adequately to 0 after $N$ samples). Take the first $N$ samples of your signal, reverse them, and prepend that to the start of your signal. In effect, we're defining a new signal $g(t)=g(-t)=f(t)$ for the purposes of filtering. When you filter that, the filter state will have already been "primed" with the initial signal: delete the first $N$ samples of the output, and the DC bump should have hopefully gone away.

Needless to say, there are numerous gotchas with that scheme &mdash; for starters, it's harder to do for realtime processing, and if the signal derivative is high at $t=0$, you are still going to have trust issues with the first few samples. But it should get rid of the bump.