If you want a clear answer, you should state precisely all the details about the situation you want to analyze.

For example, what fraction of the bonus must be gambled at each round? Depending on it, the answer will change dramatically.
Assuming, you have to gamble it entirely at each round, then the fraction of the bonus you end up after a single round is 100% - 10% = 90% = 0.9
Starting with 1BTC, you get 1 * 0.9 = 0.9BTC after one round.
The second one yields 0.9*0.9 and so on...

The formula for the EV after n consecutive plays of the totality of the bonus is: 1*0.9^n (i.e. bonus*(1 - house edge)^n) which obviously tends towards 0 as n grows to infinity.
It is never 0 for any finite n, but in practice the EV will fall under one satoshi which will be rounded as 0.