Here is my solution, everything is working as it should except the part where I used the logical OR operator.

// First question

let correctGuessLondon = false;
let score = 0;
let rank = '';
const ukCapital = prompt('What is the capital of Great Britain?');
if ( ukCapital === 'London') {
  correctGuessLondon = true;
}

if (correctGuessLondon === true) {
  score = 1;
}

// Second question

let correctGuessTrump = false;
const usPresident = prompt('Who is the president of USA?');
if ( usPresident === 'Donald Trump') {
  correctGuessTrump = true;
}

if ( correctGuessTrump === true ) {
  score += 1;
}

// Third question

let correctGuessRiver = false;
const bigRiver = prompt('What is the biggest river in Europe?');
if ( bigRiver === 'Danube' ) {
  correctGuessRiver = true;
}

if ( correctGuessRiver === true ) {
  score += 1;
}

// Forth question

let correctGuessCar = false;
const car = prompt('What is the second word in the Mercedes brand?');
if ( car === 'Benz' ) {
  correctGuessCar = true;
}

if ( correctGuessCar === true ) {
  score += 1;
}

// Fifth question

let correctGuessFingers = false;
let fingers = prompt('How many fingers has the human?');
if ( fingers === '20' || fingers === 'Twenty' ) /* THAT'S WHERE INPUT "20" 
DOESN'T WORK */ 
{
  correctGuessFingers = true;
}

if ( correctGuessFingers === true ) {
  score += 1;
}

// Player's rank and Displayed message

if ( score === 0 ) {
  rank = 'No Crown';
  document.querySelector('main').innerHTML =
  `<h1>You got 0 out of 5 questions correct.</h1>
   <p>Crown earned: ${rank}</p>`;
} else if ( score === 1 ) {
  rank = 'Bronze';
  document.querySelector('main').innerHTML =
  `<h1>You got 1 out of 5 questions correct.</h1>
   <p>Crown earned: ${rank}</p>`;
} else if ( score === 2 ) {
  rank = 'Bronze';
  document.querySelector('main').innerHTML =
  `<h1>You got 2 out of 5 questions correct.</h1>
   <p>Crown earned: ${rank}</p>`;
} else if ( score === 3 ) {
  rank = 'Silver';
  document.querySelector('main').innerHTML =
  `<h1>You got 3 out of 5 questions correct.</h1>
   <p>Crown earned: ${rank}</p>`;
} else if ( score === 4 ) {
  rank = 'Silver';
  document.querySelector('main').innerHTML =
  `<h1>You got 4 out of 5 questions correct.</h1>
   <p>Crown earned: ${rank}</p>`;
} else if ( score === 5 ) {
  rank = 'Gold';
  document.querySelector('main').innerHTML =
  `<h1>You got 5 out of 5 questions correct.</h1>
   <p>Crown earned: ${rank}</p>`;
}