You don't want to delay ever. Instead you want to change your thinking to "X miliseconds have passed, time to print and reset":

int maxval = 0;
uint32_t lastSample = 0;

void setup() {
    Serial.begin(115200);
}

void loop() {
    if (millis() - lastSample > 100) { // Every 100ms:
        lastSample = millis();
        Serial.println(maxVal);
        maxVal = 0;
    }
    int reading = analogRead(0);
    if (reading > maxVal) {
        maxVal = reading;
    }
}

Every 100ms it will print the maximum analog value read during the past 100ms.

You don't want to delay ever. Instead you want to change your thinking to "X miliseconds have passed, time to print and reset":

int maxval = 0;
uint32_t lastSample = 0;

void setup() {
    Serial.begin(115200);
}

void loop() {
    if (millis() - lastSample > 100) { // Every 100ms:
        lastSample = millis();
        Serial.println(maxVal);
        maxVal = 0;
    }
    int reading = analogRead(0);
    if (reading > maxVal) {
        maxVal = reading;
    }
}

Every 100ms it will print the maximum analog value read during the past 100ms.

You can replace the 100ms with any other value you choose, or even with a check to see if a button has been pressed, in which case you get the maximum reading since the last time the button was pressed:

// global
bool lastButtonValue = HIGH;

// in loop()
if (digitalRead(3) != lastButtonValue) {
    lastButtonValue = digitalRead(3);
    if (lastButtonValue == LOW) {
        // ... do the print and reset 
    }
}